On the unitary group $U(n)$, the Haar measure provides a suitable notion of uniformity for many applications. The Lie Group $U(n)$ is generated by the Hermitian matrices, i.e., I can write any $A \in U(n)$ as $$A = \exp(\sum_{j,k=1}^n i a_{jk}\Omega_{jk})$$ where $a_{jk}$ are real coefficients dependent on A and $\Omega_{jk}$ are the standard basis elements of the Lie Algebra $\mathfrak{u}(n) = \{X \in \mathbb{C}^{n \times n } | X^\dagger = X\}$.
My question is this: What is the measure "induced" on the Lie Algebra by the Haar measure on $U(n)$? Put another way: What measure on $\mathbb{R}^{n \times n}$ do I have to use to sample the $a_{jk}$ if I want the corresponding unitary matrices to be distributed according to the Haar measure?