I'm trying to write down the wedge product of 2 1-forms on an n-dimensional Manifold.
$\alpha = \alpha_1 dx^1 + \alpha_2 dx^2 + \cdots + \alpha_n dx^n$
$\beta = \beta_1 dx^1 + \beta_2 dx^2 + \cdots + \beta_n dx^n$
I know how to do this for the 2 and 3 dimensional case. But I'm having a problem with the n-dimensional case. More specifically, what sign the individual 2-forms get?
What I mean by this is, let's concider a few terms of $\alpha \wedge \beta$:
$\cdots \alpha_1 \beta_2 dx^1\wedge dx^2 + \alpha_2 \beta_1 dx^2 \wedge dx^1 \cdots$
Now the problem I'm having is, I think $dx^2\wedge dx^1 = -dx^1\wedge dx^2$. Which allows me to combine the above two terms. For 2 and 3 dimensions, this seems easy, as I can just look at the permutations, but in n-dimensions,
(1) how does this permutation look like? Or is it always minus if I switch the $dx$?
(2) Does this mean that $\alpha\wedge\alpha = 0$ always for 1 forms?