  We prove that if $f:\mathbb{B}^n \to \mathbb{B}^n$, for $n\geq 2$, is a
homeomorphism with bounded skew over all equilateral hyperbolic triangles, then
$f$ is in fact quasiconformal. Conversely, we show that if $f:\mathbb{B}^n \to
\mathbb{B}^n$ is quasiconformal then $f$ is $\eta$-quasisymmetric in the
hyperbolic metric, where $\eta$ depends only on $n$ and $K$. We obtain the same
result for hyperbolic $n$-manifolds. Analogous results in $\mathbb{R}^n$, and
metric spaces that behave like $\mathbb{R}^n$, are known, but as far as we are
aware, these are the first such results in the hyperbolic setting, which is the
natural metric to use on $\mathbb{B}^n$.
