In 2D euclidean space, rotation matrix is a matrix that tilts every single vector in the 2D space, without changing the scale. For example following matrix rotates the space by 90 degrees counter-clockwise.
$$R_2 = \begin{bmatrix} 0 & -1\\ 1 & 0\\ \end{bmatrix}$$
Also, the tip of each unit vector is exactly the same distance from its original position, for example above, the tip of each unit vector moved by $\sqrt{2}$. This is parameterizable and we have a family of 2D rotation matrix given by
$$R_2(\theta) = \begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{bmatrix}$$
We can say that a rotation matrix is a linear transformation that moves all points on the surface of n-dimensional ball by the same amount.
This definition works for 1D, with the matrix $R_1 = \begin{bmatrix}-1\end{bmatrix}$ or $R_1(\theta) = \begin{bmatrix}(-1)^{\theta}\end{bmatrix}$ in complex plane.
Is there a matrix $R_3$ in 3D that moves each point on the surface of the unit sphere by the same amount? Is there $R_n$ ?
In $\Bbb R^n$ with $n$ odd, each rotation (element of $SO(n)$) has $1$ as an eigenvalue, so has a fixed point on the unit sphere.
If $n$ is even, consider the block matrix $$\pmatrix{R&&&&\\&R&&&\\&&\ddots&\\&&&R}$$ where $R$ is a $2\times2$ rotation matrix. This matrix shifts each point on the unit sphere by the same distance.