The representations of the quantized enveloping algebra $U_q(\frak{sl}_n)$ are labeled by the positive weight lattice $P^+$ of the classical Lie algebra $\frak{sl}_n$. Moreover, the dual Hopf algebra $SL_q(N)$ is defined as the subspace of the Hopf dual of $U_q(\frak{sl}_n)$ consisting of the coordinate functions of these representations, where for $\lambda \in P^+$, $V(\lambda)$ the corresponding representation, $v \in V(\lambda)$, and $f \in V(\lambda)^*$, we define $$ C_{f,v}: U_q({\frak sl}_N) \to k, ~~~~~~~~ A \mapsto f(Av). $$ Now for $U_q(\frak{sl}_2)$, we have that $SL_q(2)$ is generated as an algebra by the coordinate algebra of the representation corresponding to the classical natural representation. Does this result extend to the general $U_q({\frak sl}_N)$ and $SL_q(N)$ case? In other words, is it correct to assume that the generators $u^i_j$, for the coordinate algebra of this representations are the same as the standard representations of $SL_q(N)$ Or more generally still, does it extend to all $U_q(\frak{g})$?

Consider now the representation coming from the exterior product of the natural representation with itself. I would like to know how the elements of the coordinate algebra of this representation look like in terms of the generators $u^i_j$. I can't see an easy guess that matches the dimensions.