Currently I'm studying Darling's book $[1]$ on differential forms. Now a annoying thing is bothering me. The dimension of wedge space is given by:
$$\mathrm{dim}\big[\Lambda^{p}(\mathbb{R}^{n})\big] := \frac{n!}{(n-p)!p!} \tag{1}$$
So, suppose I want to write and $2-$form in an $4-$dimensional space. Then, I need four covetors as basis $\mathcal{B} = \{\mathrm{d}x^{1},\mathrm{d}x^{2},\mathrm{d}x^{3},\mathrm{d}x^{4}\}$, to form the basis of wedge space $\mathcal{B}^{\Lambda^{2}(\mathbb{R}^{4})}$. Furthermore, the dimension of this wedge space is $6$.
Now, I wrote the $2-$form as:
$$\omega = A \mathrm{d}x^{1}\wedge\mathrm{d}x^{2} + B \mathrm{d}x^{1}\wedge\mathrm{d}x^{3} + C \mathrm{d}x^{1}\wedge\mathrm{d}x^{4} +$$
$$+D \mathrm{d}x^{2}\wedge\mathrm{d}x^{1} + E\mathrm{d}x^{2}\wedge\mathrm{d}x^{3} + F\mathrm{d}x^{2}\wedge\mathrm{d}x^{4}+$$
$$+G \mathrm{d}x^{3}\wedge\mathrm{d}x^{1} + H\mathrm{d}x^{3}\wedge\mathrm{d}x^{2} + I\mathrm{d}x^{3}\wedge\mathrm{d}x^{4}+$$
$$+J \mathrm{d}x^{4}\wedge\mathrm{d}x^{1} + K\mathrm{d}x^{4}\wedge\mathrm{d}x^{2} + L\mathrm{d}x^{4}\wedge\mathrm{d}x^{3} \tag{2}$$
But, since we have the property:
$$\mathrm{d}x^{a}\wedge\mathrm{d}x^{b} = -\mathrm{d}x^{b}\wedge\mathrm{d}x^{a}$$
I think I suppose to write the $2-$form as:
$$\omega = (A-D) \mathrm{d}x^{1}\wedge\mathrm{d}x^{2} + (B-G) \mathrm{d}x^{1}\wedge\mathrm{d}x^{3} + (C-J) \mathrm{d}x^{1}\wedge\mathrm{d}x^{4} +$$
$$+ (E-H)\mathrm{d}x^{2}\wedge\mathrm{d}x^{3} + (F-K)\mathrm{d}x^{2}\wedge\mathrm{d}x^{4}+ (I-L)\mathrm{d}x^{3}\wedge\mathrm{d}x^{4} \tag{3}$$
Now, suppose that someone gives you an form like:
$$\omega' = f \mathrm{d}x^{1}\wedge\mathrm{d}x^{2} + g \mathrm{d}x^{1}\wedge\mathrm{d}x^{3} + h \mathrm{d}x^{1}\wedge\mathrm{d}x^{4} +$$
$$+ i\mathrm{d}x^{2}\wedge\mathrm{d}x^{3} + j\mathrm{d}x^{2}\wedge\mathrm{d}x^{4}+ k\mathrm{d}x^{3}\wedge\mathrm{d}x^{4}$$
I suspect that is incorrect to state that the functions $f,g,h,i,j,k$ have the form:
$$\begin{cases}f = (a-b)\\ g = (c-d)\\ h=(e-t)\\ i=(q-u)\\j=(w-z)\\k = (r-s)\end{cases}$$ I would like to know then, how I suppose to write the $2-$form? As $(2)$ or $(3)$?
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$[1]$ DARLING.R.W.R. Differential Forms and Connections.