In differential geometry, we study the integration of differential forms, which are alternating covariant tensor fields. "Alternating" means that swapping two inputs leads to a sign change. For example, $dx\wedge dy=-dy\wedge dx$.
I understand why we need the linear form to be alternating to perform integration - if we consider Jacobian determinants of functions $U\to\mathbb R^n$ (where $U\subseteq \mathbb R^k$ is open), we will notice that as a determinant, swapping two rows leads to a sign change; so according to this, exterior forms are also expected to have this property.
However, I am still wondering if non-alternating forms could be somehow integrated. Consider the 2-form $\omega=dx\otimes dy$, defined explicitly by $\omega(a\partial_x+b\partial_y)=ab$. ($\partial_x,\partial_y$ are orthonormal basis vectors of the tangent space $T_p\mathbb R^2$.) Could we somehow make sense out of the integral $$ \int_{\partial B(0,1)} \omega \text{ ?} $$ I feel it hard to explain clearly why this would/would not be possible.