Linear independence does not really say much.
This algebraic variety is discussed in some detail in an old paper of Kirillov and Neretin https://www.mat.univie.ac.at/~neretin/kiri87a.pdf .
The case of $n=4$ which already shows many interesting phenomena (but already is nontrivial from the computer algebra viewpoint) is analysed in detail in the recent preprint https://arxiv.org/abs/2208.14631 - you might find it enlightening.