I found a reference that develops the formula in a general setting:

Integration on infinite-dimensional ... - Google Books

This comes from the book, "Integration on Infinite-Dimensional Surfaces and Its Applications," by Uglanov.

Integration on Infinite-Dimensional Surfaces and Its Applications

I've not had an opportunity to obtain the book, but from a quick glance, this appears to be the sort of derivation that I'm looking for.

For the time being, I'll leave the thread as unresolved in case someone has a different reference they've found useful.