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?
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The Lie algebra for $U(N)$ consists of $N\times N$ skew-Hermitian matrices, and the Lie algebra for $SO(N)$ consists of $N\times N$ skew-symmetric matrices. In both cases, the Lie bracket is given by the ordinary commutator $[A,B] = AB-BA$.
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.