Infinitesimal Stoke's theorem
Let $\omega$ be a 1-form and $X,Y \in \Omega^1(M)$:
$$ d w(X, Y)=X(\omega(Y))-Y(\omega(X))- \omega([X, Y]). $$Proof: see [Boothby 1986 page 224]
Idea:
Another explanation: @needham2021visual page 392.
More useful formulas here.
The general case is in Stokes' theorem.
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Author of the notes: Antonio J. Pan-Collantes
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