  The defining relations (triple relations) of n pairs of parafermion operators
f_j^\pm (j=1,...,n) are known to coincide with a set of defining relations for
the Lie algebra so(2n+1) in terms of 2n generators. With the common Hermiticity
conditions, this means that the ``parafermions of order p'' correspond to a
finite-dimensional unitary irreducible representation W(p) of so(2n+1), with
highest weight (p/2, p/2,..., p/2). Although the dimension and character of
W(p) is known by classical formulas, there is no explicit basis of W(p)
available in which the parafermion operators have a natural action. In this
paper we construct an orthogonal basis for W(p), and we present the explicit
actions of the parafermion generators on these basis vectors. We use group
theoretical techniques, in which the u(n) subalgebra of so(2n+1) plays a
crucial role: a set of Gelfand-Zetlin patterns of u(n) will be used to label
the basis vectors of W(p), and also in the explicit action (matrix elements)
certain u(n) Clebsch-Gordan coefficients are essential.
