Let $G$ be a connected Lie Group of dimesion $m<\infty$ and let $g\in G$. The Maurer-Cartan form allows us to define a map from $G$ to the space of $\mathfrak{g}$-valued forms, via 
$$g\rightarrow g^{-1}dg$$


Is this map surjective, i.e. can every $\mathfrak{g}$-valued form be written as $g^{-1}dg$ for some $g\in G$?



Does Maurer-Cartan form define surjection from Lie Group to Algebra-valued forms?
These forms are at different points $g$ of $G$ for different values of $g$, so these are not in the same cotangent space, and the question is not meaningul. If you pick only one point $g$ of $G$, you only get one $\mathfrak{g}$-valued 1-form.


