Suppose we're in $\mathbb R^n$, and we have a function on line segments ,$\omega(I)$, with values in $\mathbb R$. Give sufficient conditions for $\omega$ to be given by a generalized 1-form (that is, an $(n-1)$ -current), which is integrable (that is, can be evaluated on line segments). Obviously $\omega$ should be additive, and it should not vary too wildly as we change $I$. What is the precise statement? Is it already sufficient? 


What if we are only given a sufficiently large set of segments $\{I\}$ on which $\omega$ is defined, say those not intersecting a given hyperplane.



How to define a generalized differential form through its values on submanifolds
First of all, since you are looking for a *differential form*, $\omega$ should be a function defined on *oriented* segments. By additivity it can be extended to $1$-dimensional polygonal chains in $\newcommand{\bR}{\mathbb{R}}$ $\bR^n$. 


In his beautiful book **Geometric Integration theory**, *H. Whitney* addresses a more general problem, of characterizing which linear maps defined on $k$-dimensional polyhedral chains are representable as the integration by a $k$-form. For details I refer to Chapter V of Whitney's book. 


