This page gives the relation $\left[R\vec{\omega}\right]_{\times}=R\left[\vec{\omega}\right]_{\times}R^{T}$ where $R$ is a DCM (Direction Cosine Matrix), $\vec{v}$ is the angular velocity vector and $[\enspace]_{\times}$ represents a skew-symmetric matrix.
I'm not sure where this came from. Is it some inherent property of a skew-symmetric matrix?
The identity $[R\omega]_{\times}=R[\omega]_{\times}R^T$ can be proved based on the fact $(Ra)\times(Rb)=R(a\times b)$ which can be found in wiki.
Consider an arbitrary vector $x$, then $(R\omega)\times (Rx)=R(\omega\times x)$ and hence $[R\omega]_\times Rx=R[\omega]_\times x$. Since $x$ is arbitrary, we have $[R\omega]_\times R=R[\omega]_\times$ and hence $[R\omega]_\times =R[\omega]_\times R^T$.