What’s Differential Forms, it’s spatially varying k-form. It’s used to measure k-vectors. Superficially k-vector space looks similar/same as differential k-1 form:

Differential 1-form is a linear function mapping a 1-vector to a scalar.

Differential 2-form is to measure a 2-vector(area) to a scalar:

Reflected on math expressions, they are

Now put it to the frame of coordinates. so we can compute. First, on the base of basis vector, we can learn basis vector field, further extending, the coefficient doesnt’ have to be a constant scalar but a function of scalar, so below graph illustrate:

now compare the basis for vector field and differential 1-forms:

For example,

Another example on wedge of 1-form

The whole point of a differential 1-form is to measure vector fields.