The space time approach with E as t wedge x and B as x wedge y is purely linear algebra, not differential forms.
As opposed to the weird GA form it actually makes the physically most meaningful symmetry (Lorentz transformations) explicit. That's why it's actually used in Physics.
Anti symmetric space time tensors are the absolute standard. Further formulations that reveal other aspects, dualities, symmetries are much more niche and specialized subjects and not how the subject should be taught when first encountering it.
https://en.wikipedia.org/wiki/Covariant_formulation_of_class...
One should teach the next generation the best way possible, and not turn them into conformists.
"Standards" are things to be overcome when they've outlived their prime.
Disparaging new ideas as "niche" and "specialised" when their explicit aspiration is to be better foundations is motivated reasoning.
OK, well, MTW is a pretty standard GR textbook and it is often cited as a useful text on differential forms for math.