Eigenvalues · Hard problem

A rigid body spins so that the velocity of the point at position $\mathbf r$ is $\mathbf v = \Omega\mathbf r$, where $\Omega = \begin{pmatrix}0 & -3 & 2\\3 & 0 & -1\\-2 & 1 & 0\end{pmatrix}$ is skew-symmetric ($\Omega^T = -\Omega$). Points on the instantaneous rotation axis have zero velocity, so the axis direction is an eigenvector of $\Omega$ for eigenvalue $0$. Writing that eigenvector as $(1, b, c)$, find $b + c$.