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.
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 — you might find it enlightening.
Update: I checked the MathSciNet review of the paper of Kirillov and Neretin and found two other relevant references:
Carles, Diakité - les variétés d'Algèbres de Lie de dimension $\leqslant 7$
Gorbatsevich - Some properties of the space of n-dimensional Lie algebras (where in particular your observation on linear independence is proved)