infinitesimal generators
An infinitesimal generator is the 'seed' of a continuous symmetry: the tiny initial nudge whose repeated, exponentiated application builds the full finite transformation. For rotations the seeds are the skew-symmetric matrices; exponentiating them grows actual rotations. The generators ARE the Lie algebra, seen as the directions a symmetry can start moving.
Concretely, if gamma(t) = exp(tX) is a one-parameter subgroup, its generator is X = gamma'(0), the velocity at the identity. For SO(n) the generators are skew-symmetric (X^T = -X), because differentiating A^T A = I at t = 0 gives X^T + X = 0. For U(n) they are anti-Hermitian (X^* = -X); for SU(n), anti-Hermitian and traceless. The generator carries the same information as the whole flow it produces.
Geometrically a skew-symmetric generator encodes an axis and a rate: in 3D, the generator [0,-c,b; c,0,-a; -a-on-row... ] acts as the cross product with the vector (a,b,c), so exp(t X) rotates about that axis at unit angular speed. The familiar fact 'angular velocity is a vector' is really the statement that so(3) is 3-dimensional with this skew-symmetric basis.
Why it matters: in physics the generators ARE the conserved quantities (angular momentum generates rotations, momentum generates translations, the Hamiltonian generates time evolution) — this is Noether's theorem in matrix form. A caveat: generators do not generally commute, so [X, Y] != 0 means you cannot simply add rotations about different axes; the bracket records the leftover twist.
The skew generator X squares to -I, so its exponential collapses to a cosine-plus-sine formula — the 2x2 echo of Euler's identity.
Physics convention twist: physicists usually write a unitary as exp(i H t) with H Hermitian (the generator), so they call the HERMITIAN matrix the generator. Mathematicians use the anti-Hermitian iH. Same flow, an extra factor of i — check which an author means before chasing a sign.