As the word "form" usually denotes a mapping from a vector space into its underlying field, the more general term "multilinear map" is used, when talking about a general map that is linear in all its arguments.
For N = 2, i.e. only two variables, one calls f a bilinear form.
An important type of multilinear forms are alternating multilinear forms which have the additional property of changing their sign under exchange of two arguments. When K has characteristic other than 2, this is equivalent to saying that