= Solution
The <infinitesimal strain tensor>[infinitesimal strain] is $e_{ij}=(u_{i,j}+u_{j,i})/2$. A constant <translation> has zero <gradient>. For the remaining term, $u_i=\epsilon_{ikl}\omega_kx_l$ has $u_{i,j}=\epsilon_{ikj}\omega_k$, a <skew-symmetric matrix>. Therefore
$$
\boxed{e_{ij}=0.}
$$
Physically $\mathbf a$ is a <translation> and $\boldsymbol\omega\times\mathbf x$ is an infinitesimal <rigid motion>[rigid rotation], with axis along $\boldsymbol\omega$ and <rotation> angle $|\boldsymbol\omega|$. It is the linearized <rotation> <displacement>, not the exact <displacement> of a general finite <rotation>. This distinction matters because the <infinitesimal strain tensor>[infinitesimal strain] omits quadratic <displacement>-<gradient> terms.
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