I am trying to find the differential of the function
$$F:M=\begin{pmatrix}A & B \\ C & D\end{pmatrix}\mapsto MJM^T$$ where $J=\begin{pmatrix}0 & I \\ -I & 0\end{pmatrix}$, $A, B, C, D, 0, I$ are $n\times n$ matrices.
I got that $F(M)=\begin{pmatrix} -BA+AB & -BC+AD \\ -DA+CB & -DC+CD \end{pmatrix}$, but I have no idea how to differentiate the function with respect to matrices.