I'm running into a functional associated to a piecewise smooth curve $\gamma: [0,1] \to V$, where $V$ is a real vector space with a symplectic form $\omega$:

$$ \int_{0 \leq x \leq y \leq 1} \omega(\gamma'(x), \gamma'(y))\ dx\ dy $$

Is this a standard concept? In particular, does it have a name?

In my application, $\gamma$ is something like the path from SW to NE under a partition, and this functional is the area of the partition. (Though generally I'm working in higher dimension.)