I am searching for a book on differential forms which deals with doing plane geometry problems through forms. For instance, in beginning linear algebra, there are questions of finding equation of line and planes. These questions can also be done through differential forms. For example:
If we have a one form $a$, then if we consider the one form corresponding to the dual of the position vector of a point on the line $c$, we have the equation of line as:
$$ a \wedge c=0$$
Similarly equations can be done for lines.
Forms also allow us to treat area as a vector (in the sense of adding oriented planes). This can for instance, make questions such as finding the plane having the intersect line of two planes intersecting at an angle and perpendicular to one of them easy.
I am supposing one could solve geometrically problems using this aspect of differential forms. However, I have no book which does this. Could something be recommended?
Differential forms are really not the point. You want to work with the dual space and exterior product of vectors in a fixed vector space. I recommend you look at Dan Pedoe's Geometry: A Comprehensive Course. It has beautiful material (including classical results in projective geometry and the use of Grassmannians) and introduces exterior product of vectors very early on.