The adjoint rep for SO(m) corresponds to the young tableau:

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=> the SO(2),SO(3),SO(4) are wierd.

Let Jij = generator of rotation in xi-xj plane.
Jij=xi*d/dxj-xj*d/dxi

For SO(2) 
eijJij=J= 1 dimensional thing.

For SO(3) 1/2* eijkJij=Jk=m dimensional thing.

For SO(4) = SO(3)xSO(3) = self adjoint/anti self adjoint irreps

For SO(m) = a general irrep.

in SU(m) the adjoint rep is:

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