Consider the real projective variety on $\mathbb{R}P^n \times \mathbb{R}P^n$ defined by the bihomogeneous polynomial $x_0y_0+...+x_ny_n$, i.e. $V=\{([x_0:...:x_n],[y_0:...:y_n])|x_0y_0+...+x_ny_n=0\}$.
Since $V$ is nonsingular I can consider it as a (real) smooth manifold of dimension $n$. I am interested to know whether $V$ is orientable as a manifold or not, yet I am not sure how to check this myself.
I have tried barehand-ly constructing an orientation for each affine piece $V\cap U_i=V\cap \{x_i \neq 0\}$ and fitting them together, but to determine whether they can indeed be patched was difficult. I have also tried computing Jacobian of transition functions between the affine pieces, for if they can be made to be all positive, $V$ is orientable, however I found it difficult to modify the parametrization on each piece suitably to compare the signs.
I would appreciate if someone can provide me a way to determine the orientability of $V$. Thank you in advance.