What I call the generalized Laplace expansion of a matrix determinant is pp 21 of this material. Namely, $$\det A = \sum_{J\subset [n]} (-1)^{\sum I- \sum J} \det A_{IJ}\det A_{I’J’}$$ where $A$ is a given matrix, $I$ is a fixed subset of $[n]=\left\{1,2,\cdots,n\right\}$, $I’$ is the complement of $I$ in $[n]$, similarly for $J’$, and finally $A_{IJ}$ is a matrix composed of $A$’s rows $I$ and columns $J$.
The question is quite simple. Do you know the name of this theorem if any?
I think this theorem is necessary to define a product in the algebra of alternating functions, what we always do in the elementary differentiable manifold textbooks.
I have always known it as "Laplace expansion in multiple rows". It is called Laplace's theorem in Thomas Muir, William H. Metzler, A treatise on the theory of determinants (section 93). If you are very diligent and can read French, you may want to look up "Laplace (1772)" in Muir's History (volume 1, pp. 24--33) and check if this is actually in his work (it looks like it is, in ancient notation).