  By applying a Gr\"{o}bner-Shirshov basis of the symmetric group $S_{n}$, we
give two formulas for Schubert polynomials, either of which involves only
nonnegative monomials. We also prove some combinatorial properties of Schubert
polynomials. As applications, we give two algorithms to calculate the structure
constants for Schubert polynomials, one of which depends on Monk's formula.
