For which n≥2 there is m∈ℕ such that SO(n)≅RPᵐ ?

In the following if M and N are differentiable manifold we write M≅N iff M and N are diffeomorphic. We know that SO(2)≅S¹≅RP¹ and SO(3)≅RP³ , are this the only possibility? Well, yes, a possible argument is the following:

The universal covering of RPⁿ is Sⁿ, while that of SO(n) is Spin(n), which is an algebraic group.

Now, the only spheres that are algebraic groups are S¹, S³ and S⁷. Since the dimension of Spin(n) is n(n-1)/2, an isomorphism between Spin(n) and a sphere can occur only for n=2 and n=3.

The first case corresponds to the diffeomorphism SO(2)=RP^1, the second to the diffeomorphism SO(3)=RP^3.