The first example there is: given a base point X and two vectors V,W based at X, the 2-form gives the "signed" area of the parallelogram spanned by V and W. Determinants (which measure n-dimensional parallelograms), when viewed as functions of their column vectors, have all the properties of differential forms.
Differential forms are a bit like generalized determinants and in a sense specify a way to measure something like an abstract volume in the neighborhood of a point of a manifold, in such a way that the Jacobian needed for changing coordinates is "built in".
https://math.berkeley.edu/~wodzicki/H185.S11/podrecznik/2for...
The first example there is: given a base point X and two vectors V,W based at X, the 2-form gives the "signed" area of the parallelogram spanned by V and W. Determinants (which measure n-dimensional parallelograms), when viewed as functions of their column vectors, have all the properties of differential forms.
Differential forms are a bit like generalized determinants and in a sense specify a way to measure something like an abstract volume in the neighborhood of a point of a manifold, in such a way that the Jacobian needed for changing coordinates is "built in".