I have hard time understanding what is your notion of usual; most references indeed define the quantum linear group $M_q(N)$ by generators and relations (or by the universal property as defined by Manin) and then localize at $det_q$. So far as the construction goes. But it is of course very useful to know that one can start with the polynomial ring and then do the iterated Ore extension $N^2-1$ times to obtain $M_q(N)$. This is useful to infer many useful ring-theoretic properties. For example, it follows that the matrix bialgebra $M_q(N)$ and its Hopf envelope $GL_q(N)$ are Ore domains (the same O. Ore!), hence they are contained in the Ore quotient ring in which we can do many useful computations. Various intermediate Ore localizations are also useful.