Let $R \in SO(3)$ be a rotation matrix generates by rotating about the unit vector $w$ by $\theta$ radians. That is, $R$ satisfies $R=e^{ωˆθ}$. Note $w$^ is the skew symmetric matrix.
How would one show that the eigenvalues of $w$^ are $0$, $i$, and $-i$ , where $i = \sqrt{-1}$?
If you choose your coordinates such that $\omega=e_3$, then for the skew matrix you'll find $$\eqalign{ &\Omega = \begin{bmatrix}0&-1&0\\1&0&0\\0&0&0\end{bmatrix}\cr &\Omega e_1 = -e_2\cr &\Omega e_2 = -e_1\cr &\Omega e_3 = 0 }$$ For $k=3$ the eigenvalue is clearly zero.
For the other two basis vectors the combinations $a_\pm=(e_1\pm ie_2)\,$ satisfy $$\,{\Omega a_\pm = \pm ia_\pm}$$