I have seen a statement that complex Grassmannian of any dimension is orientable, while real Grassmannian is orientable iff it is even-dimensional. Is there a way to prove it using elementary means (e.g. not using anything more than e.g. calculations in charts, or perhaps determinant of some sort, etc.)?
There is a way to do so for real and projective spaces, which requires us to simply look at standard atlases. I am wondering if there is something similar for Grassmannian.