I would like to calculate the determinant of a $2\times2$ block skew matrix: $$ \begin{pmatrix} A & B^T \\ -B & D \\ \end{pmatrix} $$
with $A^T=-A$ and $D^T=-D$, and $A$, $B$ and $D$ are $L\times L$ square complex matrix.
Is there a fast way to calculate the determinant? By reducing the dimension from $2\times L$ to $L$?
P.S. If there is no fast way to calculate that, how about the a special case when $D=A^\dagger$ ?
$$ \begin{pmatrix} A & B^T \\ -B & A^\dagger \\ \end{pmatrix} $$
Thank you for your help.