Let $\omega=\sum_{I}C_Idx_I$, where $I=(i_1,\dots,i_n)$ is a multi-index and $C_I$ constants, be an $n$-form in $\mathbb{R}^m$, with constant coefficients. Here $dx_I$ means $dx_{i_1}\wedge\dots\wedge dx_{i_2}$.
Question: Is there a continuous and injective map $f=(f_1,\dots,f_m):\mathbb{R}^m\to\mathbb{R}^m$ such that the pullback of this form under this map $f^*\omega=\sum_{I}C_Idf_I$ equals $K dx_1\wedge\dots\wedge dx_n$, where $K$ constant? If yes what if I want my map to be linear and injective? (Here again $df_I=df_{i_1}\wedge\dots\wedge df_{i_n})$.
If the answer is no to both questions is there some additional assumption on the form that would make those true?