 SO(3) Lie algebra

Differentiating $R^{T}R=I$ gives $\mathfrak{so}(3)=\{A\in M_3(\mathbb{R}):A^T=-A\}$. The generators $L_{a}v=e_a\times v$ satisfy $[L_a,L_b]={ϵ}_{abc}{L}_c$ by the cross product identity. This identifies it with the SU(2) Lie algebra by matching a skew-Hermitian Pauli basis.