You want to think of the Haar measure $d\mu(U)$ as a way of measuring uniformity in the group $U(N)$ of unitary $N\times N$ matrices. 

To form your intuition, consider $N=1$. You then have $U=e^{i\phi}$, with $0<\phi\leq 2\pi$ and $d\mu(U)=d\phi$ measures the perimeter of the unit circle. This is a uniform measure, because $d(\phi+\phi_0)=d\phi$ for any fixed phase shift $\phi_0$. You could write the requirement of uniformity in the form $d\mu(UU_0)=d\mu(U)$, with $U_0=e^{i\phi_0}$ the unitary matrix corresponding to the phase shift $\phi_0$.

Once your intuition is formed for $N=1$, you simply generalize to $N>1$ using the same definition of uniformity, $d\mu(UU_0)=d\mu(U)$ for any fixed $U_0\in U(N)$. For orthogonal (or symplectic) matrices you use the same definition of uniformity, with $U_0$ now restricted to the orthogonal or symplectic subgroup of $U(N).$

To explicitly write down the Haar measure $d\mu(U)$ in terms of the matrix elements of $U$ is only easily done for a few small values of $N$. (In particular, there is no relationship to random directions of rows or columns, as Yemon Choi [pointed out](https://mathoverflow.net/posts/comments/194678).) You typically do not need such explicit expressions, since integrals with the Haar measure can be evaluated by using only the definition of uniformity.

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In response to the follow-up question: If you wish to evaluate Haar-measure integrals of polynomials of matrix elements of $U$, you can use the socalled [Weingarten functions](http://en.wikipedia.org/wiki/Weingarten_function).

[Puchała and Miszczak - Symbolic integration with respect to the Haar measure on the unitary group](http://arxiv.org/abs/1109.4244) gives a Mathematica program to generate these.

If you need an explicit expression for the Haar measure, the steps to take are the following:

1) parameterize your matrix $U$ in terms of a set of real parameters $\{x_i\}$.

2) calculate the metric tensor $m_{ij}$, defined by $\sum_{ij}|dU_{ij}|^2 = \sum_{ij}m_{ij}dx_i dx_j$.

3) obtain the Haar measure by equating $d\mu(U) = (\operatorname{Det} m)^{1/2}\prod_i dx_i$.

This is the general recipe. In practice, for many parameterizations the answer is in the literature. In particular, for the Haar measure in Euler angle parameterizations see:

[Tilma and Sudarshan - Generalized Euler Angle Parameterization for SU(N)](http://arxiv.org/abs/math-ph/0205016)

[Życzkowski and Kuś - Random unitary matrices](https://web.archive.org/web/20070820172003/https://www.cft.edu.pl/~karol/pdf/ZK94.pdf)