We determine the mod $p$ cohomology algebra of the infinite symmetric group for $p$ an odd prime by using what we call the Dickson characteristic classes. These classes are constructed by means of the Dickson-Mùi invariants of $GL(n,\mathbb Z_p)$ on the algebra $E(x_1,\dots,x_n)\otimes \mathbb Z_p[y_1,\dots,y_n]$. They are closely related to the classical Chern classes.