generalized "pfaffians" of non-skew matrices?

[stevengj]

@alanedelman suggested that the first formula in this paper (Hamel, 2001) could be applied to define a "Pfaffian" of an arbitrary square matrix, whether skew-symmetric or not:

Not clear if it has any useful properties for non-skew-symmetric matrices, or whether there is even a good way to compute it in general, but could be an interesting research problem to investigate.

(Note that this formula gives zero for a symmetric matrix. Also, it doesn't depend on the diagonal of the matrix at all, so if it has any useful properties they may only be for hollow matrices with zero diagonals.)