Riemannian metric on the sphere with at least one negative sectional curvature at every point

Does there exist a Riemannian metric on the $n$-sphere ($n > 2$) such that at each point some (but not every) sectional curvature is negative?

For $n=2$ it is easily seen that such a metric cannot exist.

• If you just need one example, it is quite easy to construct (and verify) an appropriate left-invariant metric on the Lie group $\mathrm{SU}(2) \cong S^3$. – Nate Eldredge Jun 19 '18 at 14:52