1- forms on a thorus

1- forms on a thorus

I think this is a very simple question but I'm not really confident in
mathematics (even if I like it very much)
Let's fix a cube $[0,1]^3$ in $R^3$ and identify opposite sides, so as to
construct a thorus T.
I'd like to construct a base for $\Omega^1(T)$ (the one forms) and to
understand which 1-forms are closed and which are exact.
1) is it possible to do that?
I have in mind that the thorus has inheritaed a 1-form $dx$ from the
embedding in $R^3$ (if we interpret the one forms as vector fields, it is
the 1-form associated to the vector field that has coordinates (1,0,0) in
$R^3$), but I'm not sure of that and I don't know how to construct this
one form in a mathematical coherent way. The fact that this thorus hasn't
got a global patch confuses me...so...
2) what is this form dx on the thorus?