Well, a (contravariant) tensor is an element of tensor product:
$$V\otimes W =: \frac{F(V\times W)}{S}$$
as exposed here: https://textosdefisica.wordpress.com/textos-de-matematica/
A antisymmetric tensor is a element of de Wedge space:
$$\Lambda (V) =: \frac{T(V)}{I}$$
as exposed here: https://en.wikipedia.org/wiki/Exterior_algebra
Futhermore, consider then the following:
As exposed in Schrödinger's book called Space-time structure (pages 14-16), the definition of the mathematical object called tensor density comes inside a problem of integration of a scalar field $A$:
$$\int A dx^{4} = \int A \left\vert \frac{\partial x^{k}}{\partial x'^{i}} \right\vert dx'^{4} \neq \int A dx'^{4}$$
Where $\Big \lvert \frac{\partial x^{k}}{\partial x'^{i}} \Big \rvert $ is the determinant of jacobian transfomation matrix $\frac{\partial x^{k}}{\partial x'^{i}}$.
Now, to the integral holds as:
$$\int A dx^{4} = \int A' dx'^{4}$$
We must consider a new transformation for scalar fields:
$$A' = \left\vert \frac{\partial x^{k}}{\partial x'^{i}} \right\vert A \tag{1}$$
So $(1)$ is the definition of a scalar density, and then we generalize to tensors in a "natural" way:
$$ D'^{ml} = \left\vert \frac{\partial x^{k}}{\partial x'^{i}} \right\vert\frac{\partial x'^{m}}{\partial x^{g}} \frac{\partial x'^{l}}{\partial x^{h}}D^{gh} \tag{2}$$
Then $(2)$ is a tensor density. My question is, how can I formalize this object with multilinear algebra? (consider only finite dimensional real vector spaces).