Definition. Pfaffian [math-000P]

Let $A$ be a skew-symmetric endomorphism of a vector space $V$ (hence $A$ can also be view as a tensor: $A \in \Lambda^2 V$) and $N = \dim{V}$ is even, the Pfaffian of $A$ is the number $\text{Pf}\ A$ defined as the constant factor in the tensor equality:

where $\{ e_1,\dots,e_N \}$ is an orthonormal basis of $V$.

The sign of Pfaffian depends on the orientation of the orthonormal basis.