  For any non-zero complex number $q$, excluding finitely many roots of unity
of small order, a linear basis for the $\mathrm{SL}(n)$ skein algebra of the
twice punctured sphere is constructed. In particular, the skein algebra is a
commutative polynomial algebra in $n-1$ generators, where each generator is
represented by an explicit $\mathrm{SL}(n)$ web, without crossings, on the
surface. This includes the case $q=1$, where the skein algebra is identified
with the coordinate ring of the $\mathrm{SL}(n)$ character variety of the twice
punctured sphere. The proof of both the spanning and linear independence
properties of the basis depends on the so-called $\mathrm{SL}(n)$ quantum trace
map, due originally to Bonahon-Wong in the case $n=2$. A consequence of the
proof is that the polynomial algebra sits as a distinguished subalgebra of the
L\^{e}-Sikora $\mathrm{SL}(n)$ stated skein algebra of the annulus. We end by
discussing the relationship with Fock-Goncharov duality.
