It seems that whenever we call something a "form": quadratic form, linear form, bilinear form, one-form, two-form, etc. it is always a linear (or perhaps not?) map from some vector space (or elsewhere?) to a field of scalars. Is this in general always true?
This is simply a question of vocabulary.
It is often, but not always the case.
For example, a modular form is not of this type (it is a map from the complex upper half-plane to itself satisfying certain conditions).