I'm interested in examples of manifolds which are orientable and such that the second Stiefel-Whitney class is a square. (Of course the second Stiefel-Whitney class should be non-zero.)

An easy example is the real projective space $RP^n$ in the case $n \equiv 1 \ (\operatorname{mod} 4)$. But unfortunately, I don't know any more examples. I know that for $2$- and $3$-manifolds have the nice property $\omega_2 = \omega_1^2$, but it implies that orientable $2$- and $3$-manifolds have always vanishing second Stiefel-Whitney class.

If we know $\omega_2 = x^2$, we can apply $\operatorname{Sq^1}$ to it to get $\operatorname{Sq^1}(\omega_2) = 0$. On the other hand, using Wu's formula and since we assume $\omega_1 = 0$, we get $\operatorname{Sq^1}(\omega_2) = \omega_3$. So this means that I'm especially looking for orientable manifolds with vanishing third Stiefel-Whitney class ... what does this condition mean geometrically?