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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'sconjecture in its strong form conjectured by Peyre for some infinite family ofvarieties of higher dimension. The varieties under consideration in this papercorrespond to the projective varieties defined by the following equation $$ x_1y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0. $$in $\mathbb{P}^{2n-1}_{\mathbb{Q}}$ for all $n \geqslant 3$. This paper comeswith an Appendix by Per Salberger.

La conjecture de Manin pour une famille de vari\'et\'es en dimension sup\'erieure