There may be an earlier source, but Adolf Hurwitz 1897 is one upper bound:

A. Hurwitz, <A HREF="https://eudml.org/doc/58378">*Über die Erzeugung der Invarianten durch Integration*,</A> Nachr. Ges. Wiss. Göttingen (1897), 71–90.

> Hurwitz’s paper introduced and
> developed the notion of an invariant measure for the matrix groups
> SO(N) and U(N). He also specified a calculus from which the explicit
> form of these measures could be computed in terms of an appropriate
> parametrisation — Hurwitz chose to use Euler angles. This enabled him
> to define and compute invariant group integrals over SO(N) and U(N).

source: <A HREF="https://arxiv.org/pdf/1512.09229.pdf">A. Hurwitz and the origins of random matrix theory in mathematics</A>