This is Exercise 4 of Section 33 of Munkres' "Analysis on Manifolds" book:
(Let $A$ be an open set in $\mathbb{R}^k$.) If $\eta$ is a $k$-form in $\mathbb{R}^k$ and if $\mathbf{a}_1,\ldots,\mathbf{a}_k$ is a basis for $\mathbb{R}^k$, then what is the (non-trivial linear) relationship between the integrals $$\int_A \eta $$ and $$\int_{\mathbf{x} \in A} \eta(\mathbf{x})\left((\mathbf{x};\mathbf{a}_1),\ldots,(\mathbf{x};\mathbf{a}_k)\right) \mathrm{?}$$ Furthermore, show that if the matrix $M$ formed by the vectors $\mathbf{a}_1,\ldots,\mathbf{a}_k$ in order as column vectors (i.e. $M$ is the matrix representation of $\mathbb{R}^k$ in the basis vectors $\mathbf{a}_1,\ldots,\mathbf{a}_k$) is special orthogonal, then the integrals above are equal.
I figured that the solution might have had something to do with change of basis matrices or with some manipulation of elementary 1-forms or something, but I'm really not sure.