  We study the operator product expansions in the chiral algebra
$\mathcal{W}_{\infty}$, first using the associativity conditions in the basis
of primary generating fields and second using a different basis coming from the
free field representation in which the OPE takes a simpler quadratic form. The
results in the quadratic basis can be compactly written using certain bilocal
combinations of the generating fields and we conjecture a closed-form formula
for the complete OPE in this basis. Next we show that the commutation relations
as well as correlation functions can be easily computed using properties of
these bilocal fields. In the last part of this paper we verify the consistency
with results derived previously by studying minimal models of
$\mathcal{W}_{\infty}$ and comparing them to known reductions of
$\mathcal{W}_{\infty}$ to $\mathcal{W}_N$. The results we obtain illustrate
nicely the role of triality symmetry in the representation theory of
$\mathcal{W}_{\infty}$.
