Let's say I like to normalise the components of a wedge product in the following way, say for two one-forms (forming a bivector):
$\omega \wedge \zeta = W_{[i} Z_{j]} dx^i \wedge dx^j = 1!1!/(1+1)! (W_iZ_j - Z_iW_j) dx^i \wedge dx^j$
Where [ ] denotes anti-symmetrisation brackets.
When I want to do an integral using exterior forms, however, let's say:
$\int f(x,y) dx \wedge dy = \int f(x(x'),y(y')) dx' \wedge dy' = \int F$
Won't my measure, embedded in the forms, become distorted by the 1/2! normalisation I put in the components of my wedge product?
I imagined something might go wrong since this factor does not feature in the determinant I would normally use, were I doing a change of variables in 'ordinary' integration.