I know that the matrix $M_{2\times 2}[\mathbb R]$ for reflecting a point about a line in $\mathbb R^2$ is:
$$ \begin{bmatrix} \cos(2\alpha) & \sin(2\alpha)\\ \sin(2\alpha) & -\cos(2\alpha) \end{bmatrix} $$
What is the matrix $M_{3\times3}[\mathbb R]$ that can perform a reflection on a vector in $\mathbb R^3$ about a line going through $(0, 0, 0)^T$?
Forget about angles for a moment and let’s work with a direction vector $\mathbf d$ of the line. You can proceed as you might for computing the reflection in a plane: decompose the vector into components parallel and orthogonal to the line, reverse the orthogonal component and reassemble. That is, $$M\mathbf v = \mathbf v_\parallel-\mathbf v_\perp = \mathbf v_\parallel-(\mathbf v - \mathbf v_\parallel) = 2\mathbf v_\parallel-\mathbf v = 2{\mathbf d^T\mathbf v \over \mathbf d^T\mathbf d}\mathbf d-\mathbf v = \left(2{\mathbf d\mathbf d^T \over \mathbf d^T\mathbf d}-I\right)\mathbf v.$$ (I’ve used a well-known formula for orthogonal projection onto a vector here.) The parenthesized expression at the end is the desired reflection matrix.
In terms of direction cosines of the line, we would have $\mathbf d=(\alpha,\beta,\gamma)^T$, so $\mathbf d^T\mathbf d=1$ and $$M = \begin{bmatrix}2\alpha^2-1 & 2\alpha\beta & 2\alpha\gamma \\ 2\alpha\beta & 2\beta^2-1 & 2\beta\gamma \\ 2\alpha\gamma & 2\beta\gamma & 2\gamma^2-1\end{bmatrix}.$$
Another approach might be to note that this reflection is equivalent to a rotation about the line through an angle of $\pi$. Applying Rodrigues’ rotation formula yields essentially the above expression.