The answer is yes. There are many places in mathematics where it would be useful to 'store' numbers/whatever in a 3-dimensional grid. That is not the problem. It's using them in a context and defining the right operations that make sense so you can combine things and do some abstract algebra.

For a specific example, start with something concrete. Consider linear transformations on the the plane, IE $\mathbb{R}^2$, using vectors $\imath = [1,0]$ and $\jmath = [0,1]$ A linear transformation from the plane to the plane can be represented by a 2 by 2 matrix. Once this is solidly understood, consider a function of **two** vector variables (again, to the plane), like $L(v_1,v_2) = w$ where $L$ is linear in both variables. This means that if you plug in a vector for either $v_1$ or $v_2$ you get a linear transformation (similarly to when you take the derivative of a function along one variable). One example might look like: $L([a_1,b_1],[a_2,b_2]) = (3a_1b_1 -5a_1b_2)[2,1] + b_2b_1[1,5]$

Now you have some coefficients involved:

$f(\imath,0) = a\imath + b\jmath$

$f(\jmath,0) = c\imath + d\jmath$

$f(0,\imath) = e\imath + f\jmath$

$f(0,\jmath) = g\imath + h\jmath$.

Notice you have eight numbers a through h here which complete describe $L$. Also, note you could arrange and label these coefficients more sensibly (how, and what are these numbers given this example?). Essentially the space of inputs is 4 dimensional, but you don't think of them as four in a row or column, but four arranged in a square. And then there is two choices for the coefficients on the output, the one for $\imath$ and the one for $\jmath$.

Now these eight numbers naturally fit in a cube, and they are essentially the matrix of $L$, called a tensor