On a smooth $n$ dimensional manifold $M$, it is possible to define a smooth scalar density as a pointwise-defined function $s$ which takes $n$ vector fields $X_1, \dots, X_n \in \Gamma(TM)$, and return a $C^\infty(M)$ function $s(X_1, \dots, X_n)$, such that for any section $A$ of $\text{End}(TM)$ (i.e. a map which associates with each point $p$ a endomorphism $A_p$ of $T_pM$ in a smooth manner),
$$ s(AX_1, \dots, AX_n) = |\det(A)| \cdot s(X_1, \dots, X_n). $$
Intuitively $s$ can be viewed as assigning a volume to each $n$ dimensional paralleliped in space. Naturally, they are the core objects which can be integrated on a manifold (differential forms allow integration on an oriented manifold). One can form a one dimensional vector bundle $\xi$ whose sections are precisely the scalar densities.
My question is whether there exists a similar construction which enables one to integrate on lower dimensional submanifolds. In particular, given a pointwise defined function $s$ which takes $k$ vector fields $X_1, \dots, X_k \in \Gamma(TM)$, and spits out a $C^\infty(M)$ function, what properties should $s$ have in order for us to consider it as a natural way of measuring the area of $k$ dimensional paralleliped's on the tangent space to $M$. One such property that should be true is that for any constants
$$ \{ a_{ij} : i,j \in \{ 1, \dots, k \} \} $$
and any vector fields $X_1, \dots X_k \in \Gamma(TM)$, we should have
$$ s \left( \sum_i a_{1i} X_i, \dots, \sum_i a_{ki} X_i \right) = |\det(a_{ij})| \cdot s(X_1, \dots, X_k). $$
Certainly, if $N$ is a $k$ dimensional submanifold, then a function $s$ with this property restricts to a scalar density for vector fields $\Gamma(TN)$, and can therefore be integrated on $N$. But this doesn't seem like the only property that is necessary, since the family of functions at each point with this property is infinite dimensional (consider the case $k = 1$ where $s$ is simply a seminorm chosen smoothly on each tangent space, and there are infinitely many seminorms). Thus it seems unlikely we can make a vector bundle whose sections are precisely the '$k$ dimensional' scalar densities. What further properties are needed?