Definition 1

Given an alternating matrix $R = (\alpha_{ij})$ $(i, j = 1, \ldots, 2m)$ of even order $2m$ over an arbitrary commutative ring $A$, the pfaffian of $R$, denoted by $\mathrm{Pf}(R)$, is the element $\mathrm{P}((\alpha_{hk}))$ of $A$, where $1 \leq h < k \leq 2m$.