I have checked Rudin's proof about Poincaré Lemma (Principles of Mathematical Analysis) and it seems to have a mistake. Through Google, I found another guy who has noted such error. More details here. And I look through the other books about differential forms, but they just talked about $C^\infty$.
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The ambiguity goes away when coefficients are taken in the basis of $\Omega^k(M)$ given by wedges $dx_{i_1}\wedge\cdots\wedge dx_{i_k}$ of ordered tuples of $1$-forms $(dx_{i_1},\ldots,dx_{i_k})$ where $i_1<\cdots<i_k$.