In an example of Stokes theorem, I found the following:
Let $\Sigma$ be a smooth manifold with dim$(\Sigma)=2$ parametrized by $x^\mu(t,\lambda)$, and let $V$ be a 1-form with $V=V_\alpha dx^\alpha$.Then
$dV=\frac{1}{2}(\partial_\mu V_\nu-\partial_\nu V_\mu)dx^\mu \wedge dx^\nu=\frac{1}{2}(\partial_\mu V_\nu-\partial_\nu V_\mu)(\frac{\partial x^\mu}{\partial t} \frac{\partial x^\nu}{\partial\lambda}-\frac{\partial x^\mu}{\partial \lambda} \frac{\partial x^\nu}{\partial t})dt\wedge d\lambda$
where in the last step we took the pullback of the 2-form.
I understand why $dV$ can be written as above, but why is $dx^\mu \wedge dx^\nu=(\frac{\partial x^\mu}{\partial t} \frac{\partial x^\nu}{\partial\lambda}-\frac{\partial x^\mu}{\partial \lambda} \frac{\partial x^\nu}{\partial t})dt\wedge d\lambda$ ? What does the pullback have to do with any of this?