Let $Σ$ be a surface with negative Euler characteristic, genus at least one and at most one boundary component. We prove that the skein algebra of $Σ$ over the field of rational functions can be algebraically generated by a finite number of simple closed curves that are naturally associated to certain generators of the mapping class group of $Σ$. The action of the mapping class group on the skein algebra gives canonical relations between these generators. From this, we conjecture a presentation for a skein algebra of $Σ$.

3 pictures, minor error in the proof of Lemma 4.3 corrected