Bilinear form vs two-form I have a basic linear algebra question since I'm confused with the definitions:
What is the difference between bilinear form and 2-form?
I looked up in Wikipedia and it says that a linear form $B(v,w)$ can be symmetric or antisymmetric. For a 2-form $\omega(X,Y)$, it has to be antisymmetric per definition. Is it the only difference?
I apologise if the question is too basic. The problem is, I don't know much about the wedge products so I would appreciate if someone could explain the 2-forms maybe with them? From what I understand, given a basis, 2-form is just a matrix?
Thank you for any hints.
 A: A multilinear k-form on an $F$-vector space $V$ is just a function from $V^k$ to $F$ which is linear in each component. These can be symmetric, alternating, or neither. Here, a bilinear form and a two-form are the same thing. In $\Bbb(R)^n$, a bilinear form can be represented by a nxn matrix, based on how it treats the basis elements. 
A differential k-form is something completely different. They can be defined as a continuous map from $\Bbb{R}^n$ to $A^k(n)$, the set of all alternating multilinear k-forms (as defined above) in $\Bbb{R}^n$. A differential 2-form would never be called a "bilinear form." 
Finally, the wedge product is a product of alternating multilinear forms (say, $k$- and $l$- forms) which returns an alternating multilinear $k+l$-form. This product satisfies associative and distributive laws with addition; if $kl$ is even, then it's commutative, and it's anti-commutative otherwise. As an example, let $\omega_1 = f \space dx + g \space dy$ and $\omega_2 = h \space dx + i \space dy$ be 1-forms in $\Bbb{R}^2$. Then
$$\omega_1 \wedge \omega_2 = fh \space dx \wedge dx + gi \space dy \wedge dy + fi \space dx \wedge dy + gh \space dy \wedge dx = (fi - gh)\space dx \wedge dy$$
which is a 2-form.
