  The symmetric group acts on polynomial differential forms on $\mathbb{R}^{n}$
through its action by permuting the coordinates. In this paper the $S_{n}%
$-invariants are shown to be freely generated by the elementary symmetric
polynomials and their exterior derivatives. A basis of the alternants in the
quotient of the ideal generated by the homogeneous invariants of positive
degree is given. In addition, the highest bigraded degrees are given for the
quotient. All of these results are consistent with predictions derived by
Garsia and Romero from a recent conjecture of Zabrocki.
