Pfaffian

Remarks

The term of $M$ does not depend on the order in which the pairs are listed, because exchanging two pairs is an even permutation. For an alternating matrix of odd size, the Pfaffian is set to $0$.

Example: for $n = 2$ there are three matchings, and $$\operatorname{pf}(A) = a_{12} a_{34} - a_{13} a_{24} + a_{14} a_{23}.$$ For the matching $\set{\set{1,3}, \set{2,4}}$, the permutation $\pi_M$ is $1 \mapsto 1, 2 \mapsto 3, 3 \mapsto 2, 4 \mapsto 4$, a transposition, so its sign is $-1$.

Non-example: the symmetric matrix $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is not alternating, because $a_{21} \ne -a_{12}$ unless $2 = 0$ in $K$, so it has no Pfaffian in this sense.

The main facts are $\operatorname{pf}(A)^2 = \det A$ and $\operatorname{pf}(B A B^T) = \det(B) \operatorname{pf}(A)$ for every $2n \times 2n$ matrix $B$, where $\det$ is the determinant. Alternating matrices over a field are the same as skew-symmetric matrices when the characteristic is not $2$.