U(N) and SO(N) are quite important groups in physics. I thought I would find this with an easy google search. Apparently NOT! What is the Lie algebra and Lie bracket of the two groups?
The answer by Owen Biesel gives the standard definition.
But if you want to see a definition in terms of generators and relations you must choose a basis and then express the commutators of that basis in terns of the basis. Usually, a Chevalley basis is used, which consists of the generators of a Cartan (= maximal commutative) subalgebra and an associated root system. See
You may wish to check check that for $U(2)$, this gives the familiar definition in terms of angular momentum.