Hello, I have a somewhat conceptual question about differential forms. I have been studying differential forms off and on for some time now and things are starting to come together for me. However, there is an irritating gap in my understanding. Regarding the geometric significance or visualization, etc. of differential forms, it is my understanding that the value of the form (the real number output) can be interpreted as the oriented size of the projection of the vector(s) on which the form acts - onto an axis. What is not clear to me is exactly which axis the vector is projected onto before the projection size is 'calculated'. What space is this axis in? What is the role of this axis? Which coordinate or variable or quantity is 'tracked' on this axis? Since the differential form is a generalization of the gradient or derivative, it seems to make sense to me that the projection would be onto an axis in the space of tangent vectors, but I cannot convince myself thoroughly of this. Can anyone offer guidance in clarifying these ideas? If so, thanks so much!