Prove that $\det\left[A^{T}B-B^{T}A\right]=\det[A+B]\cdot\det\left[A-B\right]$

Question

So I need to prove that: $$\det\left[A^{T}B-B^{T}A\right]=\det[A+B]\cdot\det\left[A-B\right]$$ where $A$, $B$ are two orthogonal matrices, but it seems I'm missing something.

Answer 1

Hint: Replace $\det(A+B)$ on the right by $\det(A^T+B^T)$ (I trust you understand why that is allowed). Now use the product formula for determinants.

You'll get a problem with signs, but notice that $A^TB-B^TA$ is skew symmetric!