I'm assuming that $R$ are the reals. In any case, you need to be able
to divide by $n!$.

Given that, the answer is **yes**. I can't locate a reference, but here's
the formula for $n=3$, say:
$$ 6 B(x,y,z) = p(x+y+z) - p(x+y) - p(y+z) - p(z+x) + p(x) + p(y) +
p(z)
$$
which should give you a hint as to the general case.

*Edit by Denis Serre*. This suggests the general formula
$$n!B(x_1,\ldots,x_n)=\sum_I(-1)^{n-|I|}p(x_I),\qquad x_I:=\sum_{i\in I}x_i.$$

*Further edit by JMF*. The formula is proved in this preprint by Erik G.F. Thomas *A polarization identity for multilinear maps*