  Let $\Free_n$ denote the free group with $n$ generators $g_1, g_2, ..., g_n$.
Let $\lambda$ stand for the left regular representation of $\Free_n$ and let
$\tau$ be the standard trace associated to $\lambda$. Given any positive
integer $d$, we study the operator space structure of the subspace
$\Word_p(n,d)$ of $L_p(\tau)$ generated by the family of operators
$\lambda(g_{i_1}g_{i_2} ... g_{i_d})$ with $1 \le i_k \le n$. Moreover, our
description of this operator space holds up to a constant which does not depend
on $n$ or $p$, so that our result remains valid for infinitely many generators.
We also consider the subspace of $L_p(\tau)$ generated by the image under
$\lambda$ of the set of reduced words of length $d$. Our result extends to any
exponent $1 \le p \le \infty$ a previous result of Buchholz for the space
$\Word_{\infty}(n,d)$. The main application is a certain interpolation theorem,
valid for any degree $d$ (extending a result of the second author restricted to
$d=1$). In the simplest case $d=2$, our theorem can be stated as follows:
consider the space $\mathcal{K}_p$ formed of all block matrices $a=(a_{ij})$
with entries in the Schatten class $S_p$, such that $a$ is in $S_p$ relative to
$\ell_2 \otimes \ell_2$ and moreover such that $(\sum_{ij} a_{ij}^* a_{ij}
)^{1/2}$ and $(\sum_{ij} a_{ij} a_{ij}^*)^{1/2}$ both belong to $S_p$. We equip
$\mathcal{K}_p$ with the maximum of the three corresponding norms. Then, for $2
\le p \le \infty$ we have $\mathcal{K}_p \simeq (\mathcal{K}_2,
\mathcal{K}_\infty)_\theta$ with $1/p = (1-\theta)/2$.
