The Wikipedia page for unitary matrices gives a general expression for the $2 \times 2$ unitary matrix:
\begin{pmatrix} a & b \\ -e^{i\phi}b^* & e^{i\phi}a^* \end{pmatrix}
with $|a^2| + |b^2| = 1$. Is there a similar general construction for the $3 \times 3$ unitary matrix?