Let $A$ and $B$ be any complex $2 \times 2$ matrices. Then a short calculation gives that
$$ \det(A \circ B) = \frac{1}{2}\left( \det(A) \operatorname{perm}(B) + \operatorname{perm}(A) \det(B) \right), $$
where $\circ$ denotes the Hadamard product and $\operatorname{perm}$ is the permanent. Is there a generalization of this formula for $\det(A \circ B)$ when $A$ and $B$ are arbitrary $n \times n$ complex matrices, with $n \geq 2$?