  Inspired by a method of La Bret\`eche relying on some unique factorisation,
we generalize work of Blomer, Br\"udern, and Salberger to prove Manin's
conjecture in its strong form conjectured by Peyre for some infinite family of
varieties of higher dimension. The varieties under consideration in this paper
correspond to the projective varieties defined by the following equation $$ x_1
y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0. $$
in $\mathbb{P}^{2n-1}_{\mathbb{Q}}$ for all $n \geqslant 3$. This paper comes
with an Appendix by Per Salberger.
