1. Symplectic Forms
Let $F : \mathbb{K}^{2n} \times \mathbb{K}^{2n} \to \mathbb{K}$ be a bilinear skew-symmetric nondegenerate form (as known as symplectic form).
Then $F(u,v) = u^TAv$ where $A = X^TJ_{2n}X$ where $X$ is invertible and
$$\begin{equation} J_{2n} = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix} \end{equation}$$
In this case, $X$ is the change of basis matrix from the standard basis to the symplectic basis that $F$ defines on $\mathbb{K}^{2n}$.
2. Symplectic Matrices
Separately, we say that a matrix $M$ is symplectic if $M^TJ_{2n}M = J_{2n}$.
What is the relation between these two concepts? They seem connected, but I wasn't able to find the relation between them.