Linear independence does not really say much. 

This algebraic variety is discussed in some detail in an old paper of Kirillov and Neretin: [The variety $A_n$ of $n$-dimensional Lie algebra structures](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 [Manivel, Sturmfels, and Sverrisdóttir - Four-Dimensional Lie Algebras Revisited](https://arxiv.org/abs/2208.14631) — you might find it enlightening.

Update: I checked the MathSciNet review of the paper of Kirillov and Neretin and found two other relevant references: 

<a href="https://www.sciencedirect.com/science/article/pii/002186938490125X">Carles, Diakité - les variétés d'Algèbres de Lie de dimension $\leqslant 7$</a>

<a href="https://iopscience.iop.org/article/10.1070/SM2009v200n02ABEH003991">Gorbatsevich - Some properties of the space of n-dimensional Lie algebras</a> (where in particular your observation on linear independence is proved)