While trying to show that $$\mathrm{SL}_2 \cong S^1 \times \mathbb{R}^2$$ as smooth manifolds, we came up with the following operation.
Suppose $V$ is an $n$-dimensional inner product space over $\mathbb{R}$. Then given any $(n-1)$-long list $L$ of linearly-independent vectors, it seems to be the case that there's a unique vector $x \in V$ such that, firstly, $x$ is orthogonal to all the vector's in the list, and secondly, $\det [L \; x] =1.$ In three dimensions, this is a bit like the cross product insofar as it's orthogonal to all the other vectors, but it's dissimilar insofar as the length of $x$ shrinks as the lengths of the vectors in $L$ grow.
Question. Does this vector $x$ have an accepted name?