linear algebra – Why $mathbb Q^n$ is not a lattice of $mathbb R^n$?

We can show that $mathbb Z^n$, additive subgroup of $mathbb R^n$, is a lattice and intuitively see that it might not be possible to generate $mathbb Q^n$ as integral multiple of $m$, $mleq n$, linearly independent vectors in $mathbb R^n$. But can you give a proof, I am having difficulty grasping the definition of lattices itself.