Just a beginner in symplectic geometry, and the definition of symplectic matrix bothers me. A $2n\times 2n$ real matrix $M$ is said to be symplectic if it satisfies the following condition: $$M^T\Omega M=\Omega$$ where $\Omega$ is a fixed $2n\times 2n$ real, invertible and skew-symmetric matrix.
My question is: since $\Omega$ can be arbitrary, so if $\Omega,\Delta$ are both satisfy the condition, then the following statement must be true: $$M^T\Omega M=\Omega \Rightarrow M^T\Delta M=\Delta.$$
But I don't know how to prove this. Can anyone help me? Thanks.