I was curious to find an argument to show orientability of the $2n$-bundle $$\gamma^n\oplus \gamma^n$$ where $\gamma^n$ is the canonical $n$-bundle over the infinite grassmannians $Gr_n(\mathbb{R}^{\infty})$. I now that using the first Stiefel-Whitney class would be easy, but I don't want to use such machinery (I'd have to introduce too many things in my work -obstruction theory I think-)
I'd prefer some precise hint rather than complete solutions, because I wanna work out details on myself, and more importantly get used to such reasonings
Hint: If $A$ is an invertible $n \times n$ real matrix and $A \oplus A$ denotes the $(2n) \times (2n)$ block matrix $\left[\begin{smallmatrix}A & 0 \\ 0 & A \\ \end{smallmatrix}\right]$, then $\det(A \oplus A) > 0$.