In a paper I've been reading ("Non-linear complementary filters on the special orthogonal group", Robert Mahony et al. link: warning PDF) there is an operation:
$P_a(\tilde{R}) = \frac{1}{2} (\tilde{R} - \tilde{R}^T)$,
where $R \in SO(3)$ and $\tilde{R}$ is an error of $R$'s estimate. In the particular case $P_a(\tilde{R})$ seems to be 'transforming' the rotation-error matrix to a skew-symmetric matrix, maybe even its derivative. Or can it really be its derivative?