Finding matrix $\mathbf{Q}$ that satisfies $\mathbf{J=Q^TAQ}$ of skew-symmetric $\mathbf{A}$ and $\mathbf{J}$

Question

Let $\mathbf{A}$ and $\mathbf{J}$ be real, full-rank, skew-symmetric, square matrices. If we know those matrices, can we find a real square matrix $\mathbf{Q}$ that satisfies $\mathbf{J=Q^TAQ}$?

Specifically, my questions are:

1. Is the existence of $\mathbf{Q}$ guaranteed?
2. If yes, is $\mathbf{Q}$ unique (i.e. only one possible solution)? If it’s not unique, how to find some of the solutions?