Certain special $k$-forms on an $n$-manifold $M$ do lend themselves to visualization.

For simplicity, let me assume that $M$ comes with a Riemannian metric.

Now suppose that we have a field $\phi$ of oriented $k$-planes and a scalar field (i.e. real-valued function) $s$ on $M$. The metric and the orientation induce $k$-forms on the $k$-planes in $\phi$. (I'll consider $k$-forms on a $k$-plane as trivial to visualize, or nearly so.)

Given a $k-tuple$ $T$ of tangent vectors at a point $p$, one can project the vectors in $T$ to $\phi(p)$, the $k$-plane over $p$, compute the volume (according to the $k$-form induced by the metric) of the resulting parallelepiped, then use the orientation of $\phi(p)$ to get a sign and finally scale by $s(p)$.

The linearity of projection makes this recipe yield an alternating multilinear function of the vectors in $T$. Thus the given data determine a $k$-form in a way roughly as easy to picture as a vector field.

General differential $k$-forms arise as linear combinations of these special ones. The usual coordinate representation of $k$-forms work exactly this way, albeit in that case the $k$-plane fields derive from the coordinate system alone, independently from the $k$-form one hopes to visualize.

That leaves the question of how much better we do (at visualization) if we adapt the $k$-plane fields to $k$-form we want to see. In general, not to much better. For large $n$, and for $k$ intermediate between $0$ and $n$, the dimension of the Grassmann variety of $k$-planes in $n$-space simply doesn't supply nearly enough degrees of freedom for significantly reducing the number of summands in the linear combination. And correct me if I err here, but I don't see anything that would make a representation with a minimal number of summands in any way canonical, even generically. But perhaps one can say something more in low dimensional cases, say $2$-forms on a $4$-manifold. A dimension count suggests two fields of $2$-planes should suffice.

Let me add that by much the reasoning above, we get the projective Grassmann coordinates on the Grassmann variet $Gr(n,k)$ of $k$-planes in $n$-space. A differential $k$-form then determines a hyperplane section of the Grassmann variety. The hyperplane section "knows" the differential form up to a scalar. (We have here an generalization of regarding a (co)vector as a magnitude and a direction.) So up to a scalar field, we can regard a differential form as a bundle of Grassmann variety hyperplane sections. That may not make it easier to "see" differential forms, but it does translate them into something geometrical, canonical and coordinate free.

Certain special $k$-forms on an $n$-manifold $M$ do lend themselves to visualization.

For simplicity, let me assume that $M$ comes with a Riemannian metric.

Now suppose that we have a field $\phi$ of oriented $k$-planes and a scalar field (i.e. real-valued function) $s$ on $M$. The metric and the orientation induce $k$-forms on the $k$-planes in $\phi$. (I'll take $k$-forms on a $k$-plane as trivial to visualize, or nearly so.)

Given a $k$-tuple $T$ of tangent vectors at a point $p$, one can project the vectors in $T$ to $\phi(p)$, the $k$-plane over $p$, compute the volume (according to the $k$-form induced by the metric) of the resulting parallelepiped, then use the orientation of $\phi(p)$ to get a sign and finally scale by $s(p)$.

The linearity of projection makes this recipe yield an alternating multilinear function of the vectors in $T$. Thus the given data determine a $k$-form in a way roughly as easy to picture as a vector field.

General differential $k$-forms arise as linear combinations of these special ones. The usual coordinate representation of $k$-forms works exactly this way, albeit in that case the $k$-plane fields derive from the coordinate system alone, independently from the $k$-form one hopes to visualize.

That leaves the question of how much better we do (at visualization) if we adapt the $k$-plane fields to the $k$-form we want to see. In general, we dod not do much better. For large $n$, and for $k$ intermediate between $0$ and $n$, the dimension of the Grassmann variety of $k$-planes in $n$-space simply doesn't supply nearly enough degrees of freedom for significantly reducing the number of summands in the linear combination. And correct me if I err here, but I don't see anything that would make a representation with a minimal number of summands in any way canonical, even generically. But perhaps one can say something more in low dimensional cases, say $2$-forms on a $4$-manifold. A dimension count suggests two fields of $2$-planes should suffice.

Let me add that by much the reasoning above, we get the projective Grassmann coordinates on the Grassmann variet $Gr(n,k)$ of $k$-planes in $n$-space. A differential $k$-form then determines a hyperplane section of the Grassmann variety. The hyperplane section "knows" the differential form up to a scalar. (We have here an generalization of regarding a (co)vector as a magnitude and a direction.) So up to a scalar field, we can regard a differential form as a bundle of Grassmann variety hyperplane sections. That may not make it easier to "see" differential forms, but it does translate them into something geometrical, canonical and coordinate free.