Given a volume form in $\mathbb{R}^3$ $$vol = x_1\wedge x_2\wedge x_3$$ then by introducing the new variables $y_{ij} = x_ix_j$ is it possible to express this as a 2-form in the $x_i$ and $y_{ij}$ coordinates?
E.g. something like $$vol = \sum_{ij} f_{ij}\, dz_{i}\wedge dz_j$$ where $z=(x_1,x_2,x_3,y_{11},y_{12},y_{13},y_{22},y_{23},y_{33})$