I wonder whether every integral lattice $\mathcal L$ is a sublattice of $\Bbb Z^n$ in the following sense:

Consider some vectors $v_1,..., v_k\in\Bbb Z^n$. Then $$\mathcal L(v_1,...,v_k):=\mathrm{span}\{v_1,...,v_k\}\cap \Bbb Z^n$$ is a lattice, and shall be called sublattice of $\Bbb Z^n$.

If not, are there some natural restriction (e.g. unimodularity) to make this true?