  F. Labourie [arXiv:1212.5015] characterized the Hitchin components for
$\operatorname{PSL}(n, \mathbb{R})$ for any $n>1$ by using the swapping
algebra, where the swapping algebra should be understood as a ring equipped
with a Poisson bracket. We introduce the rank $n$ swapping algebra, which is
the quotient of the swapping algebra by the $(n+1)\times(n+1)$ determinant
relations. The main results are the well-definedness of the rank $n$ swapping
algebra and the "cross-ratio" in its fraction algebra. As a consequence, we use
the sub fraction algebra of the rank $n$ swapping algebra generated by these
"cross-ratios" to characterize the $\operatorname{PSL}(n, \mathbb{R})$ Hitchin
component for a fixed $n>1$. We also show the relation between the rank $2$
swapping algebra and the cluster
$\mathcal{X}_{\operatorname{PGL}(2,\mathbb{R}),D_k}$-space.
