I'm going through appendix 3 of Lax's Linear Algebra book, and I'm a bit confused about the concept of the symplectic matrix. From what I understand, we are looking for a matrix $A$ which fulfills $(x,Ay)=-(Ay,x)$, where he further specifies to the matrix \begin{equation} J=\begin{pmatrix} O & I\\ -I & O \end{pmatrix} \end{equation} written in block notation. The first point of confusion is whether it is an $O$ representing an orthogonal matrix, or a zero.
My second issue is the exercise which I'm trying to work through, which is to show that any real $2n\times 2n$ anti-self-adjoint matrix $A$ can be written in the form \begin{equation} A=FJF^T \end{equation} since my approach has been to essentially define $$F^Tx=\begin{bmatrix} x_1\\ x_2 \end{bmatrix}$$ then carry out the inner product with $J$ to try and find some tautology to show that $A$ can be written in that form. However, this approach requires me to know for certain whether the element in $J$ is an $O$ or a zero. I'm also not entirely sure whether the alternating nature of the bilinear function is correct (since I think $(x,Ay)=-(y,Ax)$ would also fit the definition, but from what I've seen online it seems that the form I have written out is the correct way). Any explanation would be appreciated, but I'd still like to give working out the exercise a shot so ideally not a full answer.