  Let $(X,\left\Vert \cdot \right\Vert )$ be a real normed space of dimension
$N\in \mathbb{N}$ with a basis $(e_{i})_{1}^{N}$ such that the norm is
invariant under coordinate permutations. Assume for simplicity that the basis
constant is at most $2$. Consider any $n\in \mathbb{N}$ and $0<\varepsilon
<1/4$ such that $n\leq c(\log \varepsilon ^{-1})^{-1}\log N$. We provide an
explicit construction of a matrix that generates a $(1+\varepsilon )$ embedding
of $\ell _{2}^{n}$ into $X$.
