  We study a closed differential form on the symmetric space of positive
definite matrices, which is defined using the Pfaffian and is
$\mathsf{GL}_{2n}(\mathbb{Z})$ invariant up to a sign. It gives rise to an
infinite family of unstable classes in the compactly-supported cohomology of
the locally symmetric space for $\mathsf{GL}_{2n}(\mathbb{Z})$ with
coefficients in the orientation bundle. Furthermore, by applying the Pfaffian
forms to the dual Laplacian of graphs, and integrating them over the space of
edge lengths, we construct an infinite family of cocycles for the odd
commutative graph complex. By explicit computation, we show that the first such
cocycle gives a non-trivial class in $H^{-6}(\mathsf{GC}_3)$.
