How can I convert an action in terms of differential forms to tensors?

I have an action of a gravitational theory wich is in terms of differential forms. Now, I need to transform this action (including wedge product and exterior derivative of tetrad and metric ) to an action in terms of tensors.i mean converting it into component notation. How can I do this calculation? Or can anyone introduce me a book or paper in this way?

for example the action of a theory is

$S = \int g_{ij}(x) \partial_{\mu} X^i \, \partial^{\mu}X^j$

but it is possible to write this action as

$\bar{\delta}_XS[X,\gamma]=-\int_\Sigma \eta_{\mu \nu} \bar{\delta}X^{\mu}\wedge d \star_\gamma d X^\nu=0$

Now I want to know that how the first form can be achieved from the second one.