My question is about tensors. I have recently spent some time studying the various definitions of tensors and some tensor calculus. What I am missing now is an intuitive way to represent tensors and I really need it to understand subjects like General Relativity, so I would like to expose to you a way to understand tensor visually that I have found on the internet and ask you to check if it could be universal and well-fitting or simply wrong or not precise. Please keep in mind that I am not a mathematician but a physicist, so forgive me if I will not be rigorous.
I will begin with what I have understood about vectors and their representation.
A vector $\vec{v}$ is an absolute object, it does not depend on anything but, eventually, on time, and it can be seen as an arrow in space (3D vector for example). When you choose a basis {$\vec{e}_1, ..., \vec{e}_n$} you can represet the vector $\vec{v}$ in two different ways (which coincide if the basis is orthonormal). The first representation is obtained by counting how many vectors of the basis you have to add to obtain $\vec{v}$ (parallelogram rule), the coefficients of the sum can be represented with up indices as $v^1,...,v^n$. The second representation is obtained by taking the orthogonal projections of $\vec{v}$ on the basis vectors, these projections can be represented with low indices as $v_1,...,v_n$.
Here comes my real question: Is it correct to say that a tensor T, say a rank 2 tensor, is an absolute object and can be seen as the "union" of 2 vectors, $\vec{v}$ and $\vec{w}$, and it can be represented in 4 different ways by a matrix of which the elements are obtained taking the product of the components of $\vec{v}$ and $\vec{w}$ expressed in the covariant and contravariant form, namely $$T^{\mu\nu} = v^{\mu}\cdot w^{\nu}$$ $$T^{\mu}_{\hspace{0.3cm}\nu} = v^{\mu}\cdot w_{\nu}$$ $$T_{\mu}^{\hspace{0.2cm}\nu} = v_{\mu}\cdot w^{\nu}$$ $$T_{\mu\nu} = v_{\mu}\cdot w_{\nu}$$ So that these 4 matrices are just 4 different representations of the same object in a chosen basis, just like the covariant and contravariant representations of $\vec{v}$ are just 2 ways to see an arrow in space?
If this "model" is not correct, what are the cases in which it can hold, if there are?
I stress here that I am not looking for a formal definition of tensors but just a simple way to represent them without losing any property.