As is well known, every compact quantum group in the sense of Woronowicz has a dense Hopf $*$-sub-algebra. For the case of $q-SU(n)$ (among others) this Hopf $*$-sub-algebra is an FRT-algebra, which is to say, roughly, that they can be constructed from an R-matrix in the Yang--Baxter sense. (This goes back to the roots of quantum groups as objects in mathematical physics - the quantum inverse scattering problem to be exact.) Can we tell from the compact quantum group itself when the Hopf $*$-sub-algebra has an R-matrix structure?

Already for ${\rm SU}_q(n)$ you have to remember that there is an extra relation about the quantum determinant being $1$. Anyhow, the correspondence you mention comes the fact that intertwiners of finite dimensional representations of ${\rm SU}_q(n)$ can be constructed from eigenspace decomposition of the braiding acting on $(\mathbb{C}^n)^{\otimes k}$ (and the extra embedding $\mathbb{C} \to (\mathbb{C}^n)^{\otimes n}$ corresponding to the q-determinant). Because of the Tannaka-Krein duality, the relations between matrix coefficients correspond to intertwiners in the representation category exactly like for usual compact groups. So, if the representation category is (almost) generated by braiding, you will have an "R-matrix" structure on the regular functions. The free orthogonal groups are such examples, since they have the same representation category as ${\rm SU}_q(2)$.