The bi-invariant Haar measure on the quotient  $SL(2,\mathbb{R})/SL(2,\mathbb{Z})/SO(2,\mathbb{R})$ represents the moduli space of rank two real lattices modulo rotation and is easy to write down. 
An element of the quotient can be written (using the Isawasa decomposition) as:
$$
\left(\begin{array}{cc}
y^{1/2}&xy^{-1/2}\\
0&y^{-1/2}\\
\end{array}
\right)
$$
with $z=x+iy$ in a fundamental domain of the action of $SL(2,\mathbb{Z})$ over the upper-half plane, such as $D = \{(x,y) : x^2+y^2 \geq 1,|x| \leq 1/2,y > 0\}$.
In these coordinates, the invariant measure is a scale of the hyperbolic measure, namely
$$
 \frac{3dx dy}{\pi y^2}.
$$
Is there a similar nice parametrisation for the rank 3 case (namely explicating the bi-invariant Haar measure of $SL(3,R)/SL(3,Z)/SO(3,\mathbb{R})$?