1. When can an $n\times n$ matrix $M$ be written as a product $M=AB$, where $A^T=A$ and $B^T=-B$?

For example, a necessary condition is that the trace of $M$ vanishes. In this case, it is easy to check that for $n=2$ all traceless matrices $M$ admit such a factorization.

2. Is there a way to know all pairs of matrices $A$ and $B$ as above, for a given $M$?

3. What conditions should satisfy $M$ so that $A$ is non-singular, and what are all factorizations with $A$ non-singular?

For example, if $n=2$ and $M$ is non-singular, $A$ and $B$ are obtained easily, and are unique up to a scalar factor and non-singular. But it is possible for $A$ to be non-singular even when $M$ is singular, for example if $M=0_n$, $M=A0_n$ is a solution for any $A$.

I expect that there is literature about this kind of factorization, but I couldn't find anything.

1. When can an $n\times n$ matrix $M$ be written as a product $M=AB$, where $A^T=A$ and $B^T=-B$?

For example, a necessary condition is that the trace of $M$ vanishes. In this case, it is easy to check that for $n=2$ all traceless matrices $M$ admit such a factorization.

2. Is there a way to know all pairs of matrices $A$ and $B$ as above, for a given $M$?

3. What conditions should satisfy $M$ so that $A$ is non-singular, and what are all factorizations with $A$ non-singular?

For example, if $n=2$ and $M$ is non-singular, $A$ and $B$ are obtained easily, and are unique up to a scalar factor and non-singular. But it is possible for $A$ to be non-singular even when $M$ is singular, for example if $M=0_n$, $M=A0_n$ is a solution for any $A$.

I expect that there is literature about this kind of factorization, but I couldn't find anything.