What is (or where can I find) an *explicit* formula for the Haar measure of the group of linear symplectic transformations of $\mathbb{R}^{2n}$?

**Added 13/05/2014.**
Some clarifying remarks:

**(1)** by symplectic group I mean the group of linear transformations that preserve the canonical (symplectic) two-form in $\mathbb{R}^{2n}$.

**(2)** "explicit" is very subjective, but what I have in mind is the formula for the Haar measure of $SL(2,\mathbb{R})$ in terms of the invariant area on the hyperbolic plane (KAN decomposition seen geometrically).

**(3)** the aim is to have some intuition for the measure of certain geometrically-defined subsets of the symplectic group such as the set of all
linear symplectic transformations that send the unit ball into the ball or cylinder of radius 2. The arbitrary constant inherent in the definition of the Haar measure can be dealt with by looking at quotients of measures of geometrically-defined subsets.

**(4)** probably once I get a hold of the references Robert Bryant and Jim Humphreys have proposed I'll have no other questions (or the question will be more precise) ...