Is there a basisless definition of the suitable topology on $\Lambda^k \Bbb R^n$?

So a vector field is a section of the tangent bundle $TM \to M$ and a $k$-form is a section of $\Lambda^k T^\ast M \to M$

and they're supposed to be related

Of course there is, if $V$ is a finite dimensional vector space it's the coarsest topology so they scalar multiplication and addition are continuous, as well as every linear functional

Probably one can drop some things from our list of functions there but redundancy doesn't matter so much

ok great