We show that for any permutation w that avoids a certain set of 13 patterns of length 5 and 6, the Schubert polynomial S w can be expressed as the determinant of a matrix of elementary symmetric polynomials in a manner similar to the Jacobi–Trudi identity. For such w, this determinantal formula is equivalent to a (signed) subtraction-free expansion of S w in the basis of standard elementary monomials.