Linear Independence in Spaces of Matrices (or even tensors)

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Let's say I have a vector space $V$ of dimension $n$, and I have various elements of $V \otimes V$, which you can think of as $n \times n$ matrices. I want to check whether these elements are linearly dependent, and in some cases find a relation. But if I create a matrix space MS, it has no attribute linear_dependence, like with vector spaces.

Do I have to just create a $n^2$-dimensional vector space $W$ and then define the bilinear mapping from $V$ to $W$? And what about rank $3$ tensors, which form a $n^3$-dimensional space?