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I'm a student who is currently trying to understand differential $k$-forms on $\mathbb R^n$ as functions which assign, to each point in $\mathbb R^n$, a linear functional on the $k$-th exterior power of the tangent space to $\mathbb R^n$ at that point. What is the most clean way to define the wedge product (and/or exterior derivative) of forms in this setting, in terms of tensor powers, exterior powers, etc.?

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Find a canonical isomorphism $\Lambda^k(V)^{\ast} \cong \Lambda^k(V^{\ast})$. –  Qiaochu Yuan Jun 20 '13 at 19:57
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