Every orthogonal matrix represents a rotation around an axis

Question

Is it true that every element of the group $O(n)$ represents a rotation around some axis? I'd like this to be true in order to decompose any matrix $R \in O(n)$ as a block matrix in $O(n-1)$ and a 1 that represents a rotation around this axis.

Answer 1

This is only true for $n\le 3$. For $n\ge 4$ there are orthogonal matrices having no real eigenvalues. Such a matrix isn't "rotating" around any fixed vector. For example consider for $n=4$ matrices \begin{pmatrix} R(\theta) & 0 \\ 0 & R(\psi)\end{pmatrix} where $R(\theta)$ and $R(\psi)$ are rotation matrices, with appropriate choices of $\phi$ and $\psi$. with no real eigenvalues.