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Suppose $G$ is a $k$-dimensional Lie group of $n\times n$ matrices of the form $[g_{ij}(x_1,...,x_k)]$ where $g_{ij}$ are smooth functions. As a follow-up to the question I posted recently, I now understand (and am convinced by the proof) that the entries of

$\Omega := g^{-1} dg$

are all left-invariant 1-form. However, my book is using this method to produce left-invariant $k$-forms, by wedging all the linearly independent 1-forms together. In the first place, how am I guaranteed that $\Omega$ will contain $k$ linearly independent 1-forms? (i.e. that the entries of $\Omega$ will span the space of left-invariant 1-forms?)

(For background, you may assume that I know differential geometry at the level of Lee's smooth manifold text, chapter 1-16)

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