Can $\mathbb{Q×Q}$ be embedded in $\mathbb{R}$ as group?

Let $\{e_i: i \in I\}$ be a basis for the vector space of $\Bbb R$ over the field $\Bbb Q$. It is clear from cardinality considerations that $I$ is uncountable.

Pick any two distinct elements from the base, say $e_{i_1}$ and $e_{i_2}$. Map $(q,q') \in \Bbb Q^2$ to $qe_{i_1} + q'e_{i_2} \in \Bbb R$ and note that we have group embedding.

This of course works for any finite power of the rationals.