I know this might be a very bad/broad question, but after going through a few practice problems for finding linearly independent generators for some of the easier subgroups of $\textrm{GL}(n,F)$ [where $F$ is $\mathbb{C}$ or $\mathbb{R}$] such as $\textrm{O}(2)$, $\textrm{O}(3)$, $\textrm{U(2)}$, and $\textrm{GL}(2,\mathbb{R})$, I've noticed that although the constraints given by the specific group definition and by other motivations (i.e. Hermitian matrices) can help with this process, I can only imagine how hard it would be to "pump out" the generators for linear homogeneous transformation groups over fields of higher dimensions.
Is there a general method that is used to obtain these generators, or is there some sort of 'art' involved? (By 'art', I mean sort of like how there is an 'art' of using algebra to reduce an inequality to the triangle inequality or to a form where the Cauchy-Schwarz lemma can be employed)
Once again, sorry if this is a dumb question.