I am starting to study wedge products, and am stuck on notation. The Bachman book on differential forms says $$ \omega \wedge \nu ( v_1, v_2 ) $$ "gives the area of the parallelogram spanned by $v_1$ and $v_2$ projected on the plane containing the vectors $\omega$ and $\nu$, and multiplied by the parallelogram spanned by $\omega$ and $\nu$."
Other authors just write $$ \omega \wedge \nu $$ with no reference to a second pair of vectors $v_1$ and $v_2$.
In this second case, am I to assume that the second pair of vectors exist (what are they)? If not, are there then two different types of wedge product?
Think of the function $f_w(v) = v \cdot w$.
I can write $f_w(v)$ for some vector $v$, or I can talk about the function $f_w$. Both are legitimate objects of study, just as $\sin(\pi)$ and the sine function (as a function) are of interest.
The authors who write the second form are denoting a function that takes two vectors as arguments, but they're not writing the vectors -- they're talking about the function itself, rather than its value on a pair of vectors.