$k$-forms on $\mathbb{R}^n$

Question

Given an expression like $$ dx_1\wedge dx_2 \wedge dx_4 \left( \begin{bmatrix} 1\\2\\3\ \end{bmatrix} \ , \ \begin{bmatrix} 4\\5\\6 \end{bmatrix} \ , \ \begin{bmatrix} 7\\8\\9 \end{bmatrix} \right) \ , $$ does this make sense? With only the given information, it seems that it is implied that we have a 3-form on $\mathbb{R}^3$, in which case it does not make much sense. But why can't we just add a fourth (zero) coordinate to each of the column vectors and compute the 3-form on $\mathbb{R}^4$?

Answer 1

You are correct that the question does not make sense.