I am confident that, other than the identity matrix any $3\times3$ matrix $\mathfrak{T}$ of real numbers such that $\det \mathfrak{T}=1,$ and $\mathfrak{T}^{-1}=\mathfrak{T}^{T}$ expresses a rotation about some fixed axis. That is, every non-zero vector is actively transformed about the same axis, by the same angle. Apparently this means the matrix has exactly one unit eigenvector.
As is shown below, if we have an axle and an angle, we can produce a rotation tensor which has a matrix representation. Since it is designed to rotate a figure rigidly, we can argue from geometry that it has a unit determinant. Since, as is show, the inverse equals the transpose, and the determinant is the same for a matrix and its transpose, we can conclude algebraically that the determinant is 1.
So, if we have a rotation by a known angle about an axis, we can produce a matrix satisfying the conditions of orthogonality. How do we show that every orthogonal matrix of $SO\left(3\right)$ (except the identity matrix) represents a rotation about an axis?
Deriving the Rotation Tensor for Known Axle and Angle
If we are given an axis of rotation $\hat{\varrho}$ and an angle $\theta$ by which a vector $\mathfrak{r}_o$ is to be rotated, we can construct a rotation tensor appearing in Menzel's Mathematical Physics II-28. The figure shows the setup prior to rotating $\mathfrak{r}_o.$
Prestidigitation Identity
First we introduce the identity tensor $\mathfrak{I},$ and what I call the prestidigitation identity. The third and fourth lines justify the second. Index summation is intended.\begin{align*} \mathfrak{I}:&=\hat{\mathfrak{e}}_i \hat{\mathfrak{e}}^i\\ \hat{\varrho }\times \mathfrak{r}&=\mathfrak{I}\cdot \hat{\varrho }\times \mathfrak{r}=\mathfrak{I}\times \hat{\varrho }\cdot \mathfrak{r}\\ &=\hat{\mathfrak{e}}_i \hat{\mathfrak{e}}^i\cdot \hat{\mathfrak{e}}_j \varrho ^j\times \mathfrak{r}\\ &=\hat{\mathfrak{e}}_i \hat{\mathfrak{e}}^i\times \hat{\mathfrak{e}}_j \varrho ^j\cdot \mathfrak{r} \end{align*}
Ad Hoc Basis
Next we resolve $\mathfrak{r}$ into a component $\mathfrak{r}_{o\parallel }$ parallel to $\hat\varrho,$ and a component $\mathfrak{r}_{o\perp}$ perpendicular to $\hat\varrho.$ As an intermediate step we obtain a vector $\mathfrak{r}_{on}$ normal to the plane spanned by $\hat\varrho$ and $\mathfrak{r}_o.$
\begin{align*} \mathfrak{r}_{o\parallel }&=\hat{\varrho } \hat{\varrho }\cdot \mathfrak{r}_o\\ \mathfrak{r}_{on}&=\hat{\varrho }\times \mathfrak{r}_o\\ \mathfrak{r}_{o\perp}&=\left(\hat{\varrho }\times \mathfrak{r}_o\right)\times \hat{\varrho }\\ &=\left(\mathfrak{r}_o\hat{\varrho } -\hat{\varrho } \mathfrak{r}_o\right)\cdot \hat{\varrho } \end{align*}
Obtaining the Rotation Tensor
Using $\mathfrak{r}_{o\perp}$ and $\mathfrak{r}_{on}$ as ad hoc orthogonal (not normalized) basis vectors, we rotate $\mathfrak{r}_{o\perp}$ by $\theta$ in the plane normal to $\hat\varrho,$ and apply prestidigitation. By rearranging the resulting expression, we are able to factor out the arbitrarily chosen argument vector $\mathfrak{r}_o$ to isolate the rotation tensor $\mathfrak{R}.$
\begin{align*} \mathfrak{r}&=\mathfrak{r}_{o\parallel }+\mathfrak{r}_{o\perp}\cos (\theta )+ \mathfrak{r}_{on}\sin (\theta )\\ &=\hat{\varrho}\hat{\varrho}\cdot\mathfrak{r}_{o}+\left(\mathfrak{r}_{o}\hat{\varrho}-\hat{\varrho}\mathfrak{r}_{o}\right)\cdot\hat{\varrho}\cos(\theta)+\hat{\varrho}\times\mathfrak{r}_{o}\sin(\theta)\\&=\hat{\varrho}\hat{\varrho}\cdot\mathfrak{r}_{o}+\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)\cdot\mathfrak{r}_{o}\cos(\theta)+\mathfrak{I}\times\hat{\varrho}\cdot\mathfrak{r}_{o}\sin(\theta)\\ &=\left(\hat{\varrho}\hat{\varrho}\left(1-\cos(\theta)\right)+\mathfrak{I}\cos(\theta)+\mathfrak{I}\times\hat{\varrho}\sin(\theta)\right)\cdot\mathfrak{r}_{o}\\ &=\mathfrak{R}\cdot\mathfrak{r}_{o} \end{align*}
Symmetric $\mathfrak{S}$ and Anti-symmetric $\mathfrak{A}$ Parts of $\mathfrak{R}$
The terms of the rotation tensor involving $c=\cos(\theta)$, which we shall collectively call $\mathfrak{S}=\hat{\varrho}\hat{\varrho}\left(1-c\right)+\mathfrak{I}c,$ pertain to the axial direction of the argument vector. The term involving $s=\sin(\theta),$ which we shall call $\mathfrak{A}=\mathfrak{I}\times\hat{\varrho}s,$ relates to the oriented plane of rotation.
These tensor components exhibit symmetry and anti-symmetry in two ways. Firstly, due to the trigonometric properties of cosine (symmetric,) and sine (anti-symmetric). Secondly by transposition when expressed in matrix form. Transposing $\mathfrak{S}$ means reversing the order of the dyadic product $\hat{\varrho}\hat{\varrho}$ which is obviously symmetric. Since $\mathfrak{A}$ involves a cross product, we expect it to be anti-symmetric, but it will be useful to examine this further. Write $\rho^i$ for the direction cosines of $\hat\varrho,$ and observe that dotting with $\mathfrak{r}_o$ on the right is equivalent to multiplying the associated matrix by the column representation of $\mathfrak{r}_o$.
\begin{align*} \mathfrak{I}\times\hat{\varrho}&=\hat{\mathfrak{e}}_{i}\hat{\mathfrak{e}}_{j}\times\hat{\mathfrak{e}}_{k}\delta^{ij}\rho^{k}\\&=\left(\begin{aligned}\hat{\mathfrak{e}}_{1}\hat{\mathfrak{e}}_{1} & \times\left(\hat{\mathfrak{e}}_{2}\rho^{2}+\hat{\mathfrak{e}}_{3}\rho^{3}\right)\\ +\hat{\mathfrak{e}}_{2}\hat{\mathfrak{e}}_{2} & \times\left(\hat{\mathfrak{e}}_{3}\rho^{3}+\hat{\mathfrak{e}}_{1}\rho^{1}\right)\\ +\hat{\mathfrak{e}}_{3}\hat{\mathfrak{e}}_{3} & \times\left(\hat{\mathfrak{e}}_{1}\rho^{1}+\hat{\mathfrak{e}}_{2}\rho^{2}\right) \end{aligned} \right)\\&=\left(\begin{aligned}\hat{\mathfrak{e}}_{1} & \left(\hat{\mathfrak{e}}_{3}\rho^{2}-\hat{\mathfrak{e}}_{2}\rho^{3}\right)\\ +\hat{\mathfrak{e}}_{2} & \left(\hat{\mathfrak{e}}_{1}\rho^{3}-\hat{\mathfrak{e}}_{3}\rho^{1}\right)\\ +\hat{\mathfrak{e}}_{3} & \left(\hat{\mathfrak{e}}_{2}\rho^{1}-\hat{\mathfrak{e}}_{1}\rho^{2}\right) \end{aligned} \right)\\&=\begin{bmatrix}0 & -\rho^{3} & +\rho^{2}\\ +\rho^{3} & 0 & -\rho^{1}\\ -\rho^{2} & +\rho^{1} & 0 \end{bmatrix} \end{align*}
The Transpose of $\mathfrak{R}$ is its Inverse
It is clear from geometric reasoning that reversing the sign of $\theta$ will produce the inverse rotation tensor $\mathfrak{R}^{-1}$. Since this amounts to changing the sign of $\mathfrak{A},$ it is clear that the transpose of $\mathfrak{R}$ is its inverse.\begin{align*} \mathfrak{R}&=\mathfrak{S}+\mathfrak{A}\\ \mathfrak{R}^{-1}&=\mathfrak{S}-\mathfrak{A}\\ \mathfrak{S}^{T}&=+\mathfrak{S}\\ \mathfrak{A}^{T}&=-\mathfrak{A}\\ \mathfrak{R}^{-1}&=\mathfrak{R}^{T} \end{align*}
Using Legerdemain to Show $\mathfrak{R}\cdot\mathfrak{R}^{-1}=\mathfrak{I}$
All of this is good and well, but we should still show that our geometric argument actually produces the inverse. That is $$\mathfrak{R}\cdot\mathfrak{R}^{-1}=\left(\mathfrak{S}+\mathfrak{A}\right)\cdot\left(\mathfrak{S}-\mathfrak{A}\right)=\mathfrak{I}.$$
By linearity we have \begin{align*} \mathfrak{R}\cdot\mathfrak{R}^{-1}&=\left(\mathfrak{S}+\mathfrak{A}\right)\cdot\left(\mathfrak{S}-\mathfrak{A}\right)\\ &=\mathfrak{S}\cdot\mathfrak{S}-\mathfrak{A}\cdot\mathfrak{A}. \end{align*}
Examining the product $-\mathfrak{A}\cdot\mathfrak{A}$ first will give us the clue we need to complete the task. Computing this using the dyadic product is a bit intractable, but the matrix product is straight forward. We use the properties of direction cosines to obtain the form shown.
\begin{align*} \left(s\mathfrak{I}\times\hat{\varrho}\right)\cdot\left(-s\mathfrak{I}\times\hat{\varrho}\right) &=s^2\left[\begin{array}{ccc} 0 & -\rho^{z} & \rho^{y}\\ \rho^{z} & 0 & -\rho^{x}\\ -\rho^{y} & \rho^{x} & 0 \end{array}\right]\left[\begin{array}{ccc} 0 & \rho^{z} & -\rho^{y}\\ -\rho^{z} & 0 & \rho^{x}\\ \rho^{y} & -\rho^{x} & 0 \end{array}\right]\\ &=s^2\left[\begin{array}{ccc} 1-\rho^{x}\rho^{x} & -\rho^{x}\rho^{y} & -\rho^{x}\rho^{z}\\ -\rho^{y}\rho^{x} & 1-\rho^{y}\rho^{y} & -\rho^{y}\rho^{z}\\ -\rho^{z}\rho^{x} & -\rho^{z}\rho^{y} & 1-\rho^{z}\rho^{z} \end{array}\right]\\ &=s^2\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right) \end{align*}
We make the following observations \begin{align*} \hat{\varrho}\hat{\varrho}&=\hat{\varrho}\hat{\varrho}\cdot\hat{\varrho}\hat{\varrho} =\mathfrak{I}\cdot\hat{\varrho}\hat{\varrho}=\hat{\varrho}\hat{\varrho}\cdot\mathfrak{I}\\ 0 &=\hat{\varrho}\hat{\varrho}\cdot\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right) =\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)\cdot\hat{\varrho}\hat{\varrho}\\ \left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)\cdot\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right) &=\mathfrak{I}\cdot\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right) -\hat{\varrho}\hat{\varrho}\cdot\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)\\ &=\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)\\ \mathfrak{S}&=\hat{\varrho}\hat{\varrho}\left(1-c\right)+\mathfrak{I}c\\ &=c \left(\mathfrak{I}-\hat{\varrho } \hat{\varrho }\right)+\hat{\varrho } \hat{\varrho }\\ \mathfrak{S}\cdot\mathfrak{S}&=c^{2}\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)\cdot\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)+\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)\cdot\hat{\varrho}\hat{\varrho}+\hat{\varrho}\hat{\varrho}\cdot\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)+\hat{\varrho}\hat{\varrho}\cdot\hat{\varrho}\hat{\varrho}\\ &=c^{2}\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)+\hat{\varrho}\hat{\varrho} \end{align*}
Finally, we have the desired result.
\begin{align*} \mathfrak{R}\cdot\mathfrak{R}^{-1}&=\left(\mathfrak{S}+\mathfrak{A}\right)\cdot\left(\mathfrak{S}-\mathfrak{A}\right)\\ &=\mathfrak{S}\cdot\mathfrak{S}-\mathfrak{A}\cdot\mathfrak{A}\\ &=c^{2}\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)+\hat{\varrho}\hat{\varrho}+s^2\left(\mathfrak{I}-\hat{\varrho}\hat{\varrho}\right)\\ &=\mathfrak{I} \end{align*}
An orthogonal $3 \times 3$ matrix $A$ with determinant 1 has 1 as an eigenvalue1 Let $\mathbf w$ be a corresponding unit eigenvector. Let $\mathbf u$ be a unit vector orthogonal to $\mathbf w$. Let $\mathbf {v=w \times u}$. Then $$A\mathbf u=\cos \theta \mathbf u + \sin \theta \mathbf v,A\mathbf v=-\sin \theta \mathbf u + \cos \theta \mathbf v.$$ The angle $\theta$ is independent of the choice of $\mathbf u$.