In order to explain to non-experts what is a vectorfield, one usually describes an assignemnt of an arrow to each point of space, and this works quite well, also when moving to manifolds (where a generalized arrow will be a tangent vector). 

My question is: 
What are similar objects that can help imagining differential forms? 

Positive qualities for such object would be (for example) that it helps justifying easily change of coordinate formulas and formulas for pullbacks via functions, or that it "easily drawable", or that it helps understanding more complicated differential-form-based concepts (e.g. connections, cohomology groups, etc.).