Let $K$ be a field with $char(K) \neq 2$. Let $S$ be an invertible $n \times n$ matrix where $n>1$. Show that there exists an invertible matrix $A$ such that $A^TS+SA=0$.
I am stuck with this problem for a while. If $S$ is the identity matrix then this is just asking for a invertible skew-symmetric matrix, which obviously exists (take the one with only $1$ above and only $-1$ below the diagonal). But for arbitrary $S$ I can't find one.