Diagonalising an invertible skew symmetric matrix

[Gavroy]

I am asking for some hints to solve this exercise. Given an invertible skew symmetric matrix $A$, then show that there are invertible matrices $ R, R^T$ such that $R^T A R = \begin{pmatrix} 0 & Id \\ -Id & 0 \end{pmatrix}$, meaning that this is a block matrix that has the identity matrix in two of the four blocks and the lower one with a negative sign.

I am completely stuck!

 

[tiny-tim]

anyone have a simple method for this?