Let $R$ be a real orthogonal matrix, $$RR^T = I$$ and let $\Omega$ be a real skew-symmetric matrix, $$\Omega^T = -\Omega$$
Please show (or disprove, although I'm pretty sure it's true) that, $$ R \Omega = \Omega R$$
I.e. prove whether or not orthogonal matrices and skew-symmetric matrices always commute in multiplication.
Is it possible to show using only the defining properties I listed? Or perhaps it might be necessary to also use the fact that skew-symmetric matrices commute with their transposes.