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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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If you have a copy of John Lee's book on Smooth manifolds, you can find the important details of these computations in chapter 8. I don't have my book with me right now, but I'll update with specific page information tomorrow. –  jmracek Oct 9 '12 at 3:55

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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 $[A,B] = AB-BA$.

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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
http://en.wikipedia.org/wiki/Chevalley_basis

You may wish to check check that for $U(2)$, this gives the familiar definition in terms of angular momentum.

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