How to sketch a differential form

Question

I deal with differential geometry for a while now but still don't have a intuitional graphical understanding for differential forms, more precisely 1-forms.

For example, how to sketch the 1-form $dx$?

Also, I don't understand this example from wikipedia:

https://en.wikipedia.org/wiki/Contact_geometry#/media/File:Standard_contact_structure.svg

I try different approaches to understand this,

1. using the curve-definition of tangent vectors

2. using the definition as maps $C^{\infty}(M) \rightarrow \mathbb{R}$

but I still don't understand.

Can somebody help me here and give a hint, which definition is better to use to understand sketches of 1-forms?