My book defines differential forms of degree $r$ in an open $U\subset \mathbb{R}^m$ as being an application $w:U\to\mathbb A_r(\mathbb{R}^m)$, where $A_r(\mathbb{R}^m)$ is the set of $r$-linear alternating applications from $\mathbb{R}^m$ to $\mathbb{R}$
So, why is $w(x,y,z) = f(x,y,z) dx \wedge dy + g(x,y,z) dx \wedge dz + h(x,y,z) dy\wedge dz$ a $2$-form? Why is it $2$-linear? Why is alternating?
As I see, $w$ would have to be linear in $2$ coordinates and not in the one left. I'm lost, could somebody help?