Let $X$ and $Y$ be $n \times n$ anti-Hermitian matrices. Then $X^{\star}_{jk} = - X_{kj}$.
I'm trying to evaluate the following sum: $$ S = \sum_{j=1}^{n} \left[ \frac{1}{j} \sum_{k=1}^{j-1} \big( Y_{jk} X_{kj} + X_{jk} Y_{kj} \big) - \frac{1}{j} \sum_{k=j+1}^{n} \big( Y_{jk} X_{kj} + X_{jk} Y_{kj} \big) \right] $$
I am pretty sure that this sum should go to zero, but I have no idea how I can show this. The $\frac{1}{j}$ factors are what really throw me off.
Are there any tricks I can use to evaluate this sum?
Ideally I wouldn't even have to use the condition that the matrices are anti-Hermitian, but I think that I might have to.