It is well-known that any real anti-symmetric $n \times n$ matrix $A$ can be transformed via

$A \to O A O^T$ into block-diagonal form consisting of $2 \times 2$ antisymmetric matrices,
where $O \in SO(n)$ is orthogonal.

It seems that the analogous statement should also hold for $O \in SO(p,n-p)$, or at least for $SO(1,n-1)$. What is the precise statement, and where can one find it?