Can anyone clarify the definition of orientation forms on a smooth manifold $M^n$?
I'm reading Lee's book "Introduction to smooth manifolds" (second edition) on pg. 381, and I am confused about how I verify that a differential $n$-form on $M$ is a form of orientation.
Is an $n$-form $\omega$ an orientation form when $\omega_p$ is not a null $n$-covector for all $p\in M$ or
$\omega_p(v_1,...,v_n)\neq0$ for all $\{v_1,...,v_n\}$ basis of $T_p M$?
As I already indicated, any nowhere-vanishing $n$-form on an $n$-dimensional manifold can be called an orientation form for the manifold. (The rationale for this is what I already commented: such an $n$-form allows you to test whether any basis for the tangent space is "positive" or "negative"; local consistency follows immediately.)
Now, perhaps you want to understand more clearly what a nowhere-vanishing $n$-form $\omega$ is. Here are two alternative explanations. The first is what you quoted from Jack Lee: For every point $p\in M$ and any basis $\{v_1,\dots,v_n\}$ for $T_pM$, the value $\omega_p(v_1,\dots,v_n)$ is nonzero. The second is to write $\omega = f(x^1,\dots,x^n)dx^1\wedge\dots\wedge dx^n$ in any local coordinate system and require that the function $f$ be nowhere-zero. (You can check that this criterion holds independent of the coordinate system.)