  We find a closed formula for the triple integral on spheres in
$\mathbb{R}^{2n}\times\mathbb{R}^{2n}\times\mathbb{R}^{2n}$ whose kernel is
given by powers of the standard symplectic form. This gives a new proof to the
Bernstein--Reznikov integral formula in the $n=1$ case. Our method also applies
for linear and conformal structures.
