You have the history backwards. Differential forms came first; and the general integrability theorem actually preceded differential forms, going back to Clairaut, 1739-1740.

For an equation of the form $A dx + B dy = dC$, Clairaut used Taylor expansions to prove the necessity of $∂A/∂y = ∂B/∂x$ and indefinite integrals to prove sufficiency. Cauchy used definite integrals (1823): $C(x,y) = \int_0^x A(X,y) dX + \int_0^y B(0,Y) dY + C(0,0)$ for sufficiency. It might be possible to use the 2nd order Taylor's Theorem with remainder to *directly* establish necessity, eliminating the need to use any infinite series expansion. This requires only $C^2$-ness, which is what is already required for the theorem.

The exterior algebra (that is: where differentials anti-commute with each other) is from Grassmann in the 1840's. Before Maxwell stripped down Hamilton's quaternions to a vector algebra and applied it in his treatise in the 1870's, he used differential forms in his papers in the 1850's and 1860's; and also made much more use of them in the treatise than he did vectors. But he only made sparing use of Grassmann's anti-commuting rule, just one place in the treatise, as far as I know.

Here: this will clear things up. This is what you were really asking and looking for. https://sites.math.washington.edu/~morrow/335_17/diff%20forms%20history%20katz.pdf