Write explicitly the (2 times 2) matrices (g=e^{i alpha_{i} sigma_{i} / 2}), where (sigma_{i}) are the Pauli

Question:

Write explicitly the $2 \times 2$ matrices $g=e^{i \alpha_{i} \sigma_{i} / 2}$, where $\sigma_{i}$ are the Pauli matrices, and compare with $g(\theta, \phi, \psi)$ for $S O(3)$, to find an explicit map between $\alpha_{i}$ and the Euler angles $(\theta, \phi, \psi)$.

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