The bilinear KP hierarchy was first described in terms of Plücker relations on an infinite dimensional Grassmann manifold. This paper describes a construction of the same hierarchy exploiting the fact that these equations are satisfied by a class of wronskian determinants. In this construction Hirota derivatives are identified with supersymmetric polynomials and the proof of the main result uses a criterion for the supersymmetry of doubly symmetric polynomials [see J. R. Stembridge, J. Algebra 95, 439–444 (1985; Zbl 0573.17004)].