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.
