In this paper we prove the existence of a complete cap of ${\rm PG}(4n+1, q)$ of size $2(q^{2n+1}-1)/(q-1)$, for each prime power $q>2$. It is obtained by projecting two disjoint Veronese varieties of ${\rm PG}(2n^2+3n, q)$ from a suitable $(2n^2-n-2)$-dimensional projective space. This shows that the trivial lower bound for the size of the smallest complete cap of ${\rm PG}(4n+1, q)$ is essentially sharp.