In this short note we present an elementary matrix-constructive algorithmic proof of the Quillen-Suslin theorem for Ore extensions A := K[ x; σ, δ], where K is a division ring, σ : K → K is a division ring automorphism and σ : K → K is a σ-derivation of K. It asserts that every finitely generated projective A-module is free. We construct a symbolic algorithm that computes the basis of a given finitely generated projective A-module. The algorithm is implemented in a computational package. Its efficiency is illustrated by four representative examples.