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. 

**(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) ...