It is well-known that there are some manifolds which are not oriented (at least for the one I know is Mobius).
Moreover, we have a criterion which asserts that: any nonvanishing m-form $w$ on $M$ determines a unique orientation of $M$ for which $w$ is positively oriented at each point. (M is smooth m-dimensional manifold).
From that, if we define the form as follows :$w=1 dx_{1}\wedge dx_{2}\ldots\wedge dx_{m} $ for m-dimensional manifold smooth M, then the manifold will be oriented. (cause the form is not vanish).
So, please could you let me know that what is wrong here?.
I think that I have some problems with differential forms, I am still not able to distinguish the differences between the differential forms (the same for vector fields) of those-different manifolds. Because in locally, when we are dealing with computations, it is look like the same (formulations).
Your definition of $\omega$ only makes sense locally. So when you write $dx_1 \wedge ... \wedge dx_m$ you are working in local coordinates on your manifold. However, usually your manifold won't have a single chart that covers all of it. Now it is not clear (and not always true..), that you can "glue" those differential forms you've got locally together to form a global differential form.