Given is a second-order tensor $T$, and three arbitrary vectors, $u$, $v$ and $w$, defined in Euclidean point space $\mathcal{E}$.
Prove that the determinant of the tensor $T$
$\det T=\frac{Tu.(Tv \times Tw)}{u.(v \times w)}$
is independent of $u$, $v$ and $w$, as long as they form a base in the 3-dimensional vector space $\mathcal{V}$ defined on $\mathcal{E}$ (that is, vectors $u$, $v$ and $w$ are linearly independent).
This is basically proving that the determinant of a tensor is invariant of the particular basis it happens to be resolved in.
I've tried to work through this using index notation, but it got pretty cumbersome quickly, so I thought there might be another way to do it. I'm not looking for the full treatment here, but rather as a general advice how to tackle this. I think I can manage the details myself.