I would like to compute the 3 × 3 Jacobian of
$$ \log(R \exp(m)) $$
with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is the inverse of the exponential map, and $R$ is a constant 3 × 3 rotation matrix.
I understand that the inverse of the exponential is not well-defined everywhere. Is there an expression for this Jacobian that holds almost everywhere?