Simple expression for Haar measure of $U(N)$ and $SU(n)$

Question

The Haar measure for the group of real $n \times n$ invertible matrices has a simple expression:

$$\mathrm d\mu=\dfrac{\mathrm dA}{|\det(A)|^n}\qquad\color{blue}{(1)}$$

The invariance under group action is immediate once one considers the Jacobian.

I want to know:

1. Is there also a simple expression for the Haar measure of the groups $U(N)$ and $SU(N)$? I found discussions like this, but they describe parameterizations of the measure, and I do not understand how these parametrizations are derived from expressions equivalent to $(1)$.
2. Why can't $(1)$ work also for $U(N)$ and $SU(N)$? Or maybe something like $\mathrm{Tr}\{ U^\dagger dU \}$?

I am a physicist who works on quantum information, and my understanding of measure theory is not so deep and very intuitive. I'd appreciate an answer in simple terms, as much as possible.