A $2n\times 2n$ matrix $A$ is called symplectic if $A^T J A = J$, where $J$ is a fixed invertible, skew symmetric matrix. Generally, $J$ is taken to be the block matrix $J = \begin{pmatrix} 0 & I_n \\ -I_n & 0\end{pmatrix}$.
Is the notion of symplectic matrix independent of the choice of $J$?