Note that this is a partial duplicate of my math.stackexchange question here. In this post I am asking something slightly broader. Note that I am a mathematical physicist and not a representation theorist / Lie algebrist / Lie group researcher so I apologise for any mistakes and would gratefully accept any corrections!
It is well known that the group law for a Lie group is locally (i.e. in some neighborhood around the identity) encoded in the Lie algebra. This encoding can be expressed by the Baker-Campbell-Haussdorf-Dynkin formula and the fact that the BCH formula does not generally converge essentially expresses the failure of the Lie algebra to capture the group law globally. However I do not know anything about where/whether the BCH formula converges for the special orthogonal group.
Because of some physics problem that would be off-topic for this forum I am interested in expressing the product of two $\mathrm{SO}(2n,R)$ matrices exclusively in terms of the Lie algebra. In other words I want a procedure that will take two $ 2n\times 2n $ real, skew-symmetric matrices $A$ and $B$ (elements of $\mathfrak{so}(2n,R))$ and find a third $C$ such that
\begin{align} e^A e^B = e^C. \end{align}
My intuition is that if $C$ is going to be obtained from $A$ and $B$ in any way it is probably going to require BCH in some form, but I would be very happy if I was wrong about that.
I expect that a sensible "first step" would be to first block-diagonalise $A$ and $B$ and use periodicity to make the eigenvalues small (in magnitude). Doing this it is possible (WLOG) to assume $A$ and $B$ have eigenvalues in the ingerval $[-i\pi, i\pi)$, for example.
