We initiate a parametric study of families of polynomial skew products, i.e., polynomial endomorphisms of C 2 of the form F (z, w) = (p(z), q(z, w)) that extend to endomorphisms of P 2 (C). Our aim is to study and give a precise characterization of the bifurcation current and the bifurcation locus of such a family. As an application, we precisely describe the geometry of the bifurcation current near infinity, and give a classification of the hyperbolic components. This is the first study of a bifurcation locus and current for an explicit and somehow general family in dimension larger than 1.