Closed-form for eigenvectors of rotation matrix

Question

For matrices that are elements of $SO(3)$ is there a formula for the eigenvectors corresponding to the eigenvalue $1$ in terms of the entries of the matrix?

Answer 1

Let $A \in SO(3)$. The matrix $A-A^T$ is skew-symmetric, hence of the form $$ \begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix} $$ for some $a$, $b$, $c$. Then the vector $(-c,b,-a)^T$ is an eigenvector of $A$ with eigenvalue $1$.

This works since if $F$ is a rotation by the angle $\phi$ around the vector $n$, then the transformation $F-F^{-1}$ geometrically equals $\sin\phi$ times the operation of taking the cross product with $n$.