Positive-dimensional real projective space is orientable exactly in odd dimension

Fix $x\in S^n$ and a positive tangent basis $(v_1,\dots ,v_n)$ at $x$. By [L1], $(x,v_1,\dots ,v_n)$ is positive in ${\mathbb{R}}^{n+1}$. Since $d{a}_x(v_i)=-v_i$, the corresponding ambient tuple at $-x$ is $(-x,-v_1,\dots ,-v_n)$, whose sign relative to the original tuple is $(-1{)}^{n+1}$. Thus [L2] gives the antipodal map the constant orientation sign $(-1{)}^{n+1}$.

If $a$ preserves orientation, define the orientation ray at $[x]$ by pushing the ray at $x$ forward with $d{\pi }_x$. The other lift is $a(x)$, and $\pi \circ a=\pi$ makes the resulting ray independent of that choice. Conversely, an orientation on ${\mathbb{R}\mathbb{P}}^n$ pulls back through the local diffeomorphism $\pi$ to an orientation of $S^n$ that $a$ must preserve. By step 1.1 this occurs exactly when $(-1{)}^{n+1}=1$, namely when $n$ is odd. Finally, ${\mathbb{R}\mathbb{P}}^0$ is a point and is orientable by [L3].