Determinant as a sum of determinants of submatrices

Question

I have seen once a formula for calculating the determinant of a block matrix which was something like this. Suppose $M$ is a matrix given by: $$M = \begin{pmatrix} 0 & A \\ -B & 0 \end{pmatrix} $$ where $A$ and $B$ are both $n \times n$ ($n$ even) positive matrices such that $\det M \neq 0$. For $I,J \subseteq \{1,...,n\}$, $A_{I\times J} = (a_{ij})_{i\in I, j\in J}$ is the restriction of the matrix $A$ with respect to the sets $I,J$. If my memory serves me right, the formula I mentioned was something like: $$\det M_{K \times K} = \sum_{I, J \subseteq K}\det A_{I\times J}\det B_{K\setminus I \times K\setminus J}$$ Maybe the above formula has some sign errors, but I would like to know if this formula (or some variant of it, where the determinant of $M$ reduces to the sum of determinants of submatrices indexed by partitions of $K$) exists indeed and how to prove it. Thanks in advance!