If $V$ is a finite-dimensional vector space, we have some $A \in \operatorname{End}(V)$ then we can extend $A$ to act on the exterior algebra $\bigwedge(A)$ by setting $$A \cdot (v_1 \wedge \cdots \wedge v_k) := (A v_1) \wedge \cdots \wedge (A v_k).$$ If $V$ is $n$-dimensional then $\bigwedge\nolimits^n V$ is one-dimensional, so the action of $A$ on this space is just scalar multiplication. Of course, we call this scalar the determinant.
If we have a skew-symmetric bilinear form $Q$ on $V$, we can regard it as an element of $(V \wedge V)^\ast$, and given the superficial similarity between the definitions of the Pfaffian and the determinant it seems as though we should be able to give a definition of the Pfaffian that involves lifting $Q$ to some larger algebra, maybe $\bigoplus_i \bigwedge\nolimits^{2i} V \subseteq \bigwedge(V)$. But I can't see a reasonable way of doing this.