  We show that a generic real projective $n$-dimensional hypersurface of odd
degree $d$, such that $4(n-2)=\binom{d+3}3$, contains "many" real 3-planes,
namely, in the logarithmic scale their number has the same rate of growth,
$d^3\log d$, as the number of complex 3-planes. This estimate is based on the
interpretation of a suitable signed count of the 3-planes as the Euler number
of an appropriate bundle.
