Quite in general, let  M be an  R-module represented as the cockernel of a linear map f: G\to F of free R-moduels F and G of rank n and m. Then Sym(M) is isomorphic (as an R-algebra) to Sym(F)/J where J is the ideal of Sym(F) generated by the image of f (the syzygies of M). Choosing basis  Sym(F) is isomorphic to R[x_1,\dots,x_n] and J is generated by m elements of degree 1 that correspond to the syzygies of M. 

These are well known properties of the symmetric algebra, see for example Bourbaki Algebra I: Chapters 1-3.