The form of an N $\times$ N unitary matrix. I recently came across this fact that any $ 2 \times 2 $ unitary matrix can be expressed as $ \begin{bmatrix}
w & z\\
-\overline{z}e^{i\theta} & \overline{w}e^{i\theta}
\end{bmatrix}$ for some $\theta \in \mathbb{R}$ and $w,z \in \mathbb{C}$. I wanted to ask if there is any similar general form for $N\times N$ unitary matrices too.
 A: Any unitary matrix $\bf{U}$ can be written as $\exp(i\bf{H})$ where $\bf{H}$ is Hermitian. A brief explanation, where I use $\mathbf{A}^*$ to denote the conjugate transpose of a matrix $\bf{A}$.
$$\mathbf{U}\mathbf{U}^*=\mathbf{I}\\\exp(i\mathbf{H})\times\exp(i\mathbf{H})^*=\exp(i\mathbf{H})\times\exp(-i\mathbf{H}^*)=\exp(i\mathbf{H})\times\exp(-i\mathbf{H})\\\exp(i\mathbf{H})\times\exp(-i\mathbf{H})=\exp(i\mathbf{H}-i\mathbf{H})=\exp(\mathbf{0})=\mathbf{I}\\\therefore\mathbf{U}\mathbf{U}^*=\exp(i\mathbf{H})\times\exp(i\mathbf{H})^*\implies\exists\mathbf{H}:\mathbf{U}=\exp(i\mathbf{H})$$
Now in the last step, the "implies" part, this is a weak step: I don't think this is at all a rigorous proof, but I'm just trying to show that it can be true. According to Wikipedia, it is in fact true for every single unitary matrix.
The conjugate transpose of $\bf{H}$ is by definition also just $\bf{H}$, explaining one step, and the first step is true since in the expansion of the matrix exponential it is straightforward to see that $\exp(i\bf{H})^*$ is the same as $\exp(-i\bf{H})$ by some properties of the complex conjugate e.g. $\overline{a+b}=\overline{a}+\overline{b}$.
VERY importantly, note that I can only put the multiplication of two exponentials as one exponential in the second line since $i\bf{H}$ and $-i\bf{H}$ commute with one another. In general, $\exp(\bf A+B)\neq\exp(A)\times\exp(B)$ when $\bf A,B$ don't commute.
