Show that $\mathbb RP^n$ with the standard smooth structure is orientable if and only if $n$ is odd.
I tried to show that the Jacobian of the transition map $$(u_1,u_2,\dots ,u_n)\mapsto \left(\frac{u_1}{u_i},\dots ,\underbrace{\frac{1}{u_i}}_{j-\text{th}},\dots ,\frac{u_n}{u_i}\right)$$ is positive only when $n$ is odd.
But, the sign of the Jacobian actually depends on whether $i$ is less than or greater than $j$.
Is there any other map that I can use?
Edit: I don't know why someone would close this question as a duplicate of this. My question is clearly much more general- the Jacobian of the general map is a much more complicated object, and the given link does not provide any insight on why it is an if and only if condition.