By a _pairing_ on a vector space $V$, I mean a linear map $A : V \otimes V \to R$.  If $V$ is $n$-dimensional ($n < \infty$), then I can define the _determinant_ of $A$ by considering the canonical action of $A$ on $\bigwedge^n V \otimes \bigwedge^n V \to R$.  In fact, I should _define_ $\det A$ to be this map, which I will call $\bigwedge^n A$.  Remember that $\bigwedge^n V$ is one-dimensional, and pairs canonically with the space of volume forms on $V$.  So if $V$ has a volume form $\mathrm{vol}$, we can define $\det A$ as $\bigwedge^n A(\mathrm{vol}^{-1} \otimes \mathrm{vol}^{-1})$.

For comparison, if I had an operator $A: V\to V$, then I would look at the map $\bigwedge^n A: \bigwedge^n V \to \bigwedge^n V$.  But $\bigwedge^n$ is one-dimensional, so $\bigwedge^n A$ consists of multiplication by a scalar, and this scalar is $\det A$.  On the other hand, if you pick an identification $V \cong V^*$, then you can think of a pairing as an operator; the identification determines a non-zero volume form, and the notions of determinants are the same.

In any case, these definitions don't work for infinite-dimensional vector spaces, because there's no ``wedge-top''.  I'd like a notion of "determinant of a pairing" like the <a href="http://mathematics.stackexchange.com/questions/183/zeta-function-regularization-of-determinants-and-traces">zeta-function regularized determinant of an operator</a>.