Let $\mathbb{H}^n$ be the quotient $GL_n(\mathbb{R})/O(n) \cdot \mathbb{R}^\times$, which at least in the theory of automorphic forms is called a generalized upper half plane. We could also think of $SL_n(\mathbb{R})/SO(n)$. This space has an invariant Riemannian metric and therefore also an induced invariant distance function, say $\rho(z,w)$.
In the case $n=2$, we can describe the distance function quite explicitly if we think of points in $\mathbb{H}^2$ as complex numbers. Among various formulas we have $$\rho(z,w) = \log \frac{|z-\bar{w}|+|z-w|}{|z-\bar{w}|-|z-w|},$$ or the very useful $\cosh \rho(z,w) = 1+2u(z,w)$, where $$u(z,w) = \frac{|z-w|^2}{4 \Im z \Im w}.$$ The function $u$ is usually easier to handle and the preferred substitute for $\rho$. Going towards the general case and thinking about points of $\mathbb{H}^2$ as matrices, say $$z= \begin{pmatrix} y & x \\ 0 & 1 \end{pmatrix}$$ corresponding to the complex number $x + iy$, then we can write $$ \cosh \rho(z,w) = \frac{\| w^{-1}z \|^2_F}{2 \det(w^{-1}z)},$$ where $\|\cdot\|_F$ is the Frobenius norm.
I am wondering if there is any way in which we could make the distance function $\rho$ on $\mathbb{H}^n$ (almost) as explicit as the formulas above. This is a rather vague question, so perhaps a better question is: what is the precise relation between $\rho$ and the Frobenius norm for $n>2$? Would another bi-$K$-invariant norm be more appropriate (but then not as easy to compute explicitly...)?
It seems to me that the case $n=2$ has many advantages, e.g. the complex numbers and the easy parametrisation of $SO(2)$. But perhaps there are some explicit formulas (so ideally not using integrals) for $n>2$ too, albeit more complicated.
