For questions about Pfaffians of skew-symmetric matrices, $\det(A)=\operatorname{pf}(A)^2$.
The determinant of a skew-symmetric matrix is a square of some polynomial in coefficients of the matrix, $\det(A)=\operatorname{pf}(A)^2$. This polynomial is called Pfaffian.
Explicitly $$ \operatorname{pf}(A)= \frac{1}{2^n n!}\sum_{\sigma \in S_{2n}}\operatorname{sgn}(\sigma) \prod_{i = 1}^n A_{\sigma(2i-1),\sigma(2i)} $$
Pfaffians are applied for counting perfect matching in graphs (domino tilings &c).