I don't know the proper terms for what I'm talking about, so I hope you'll forgive my clumsy explanation of what I want to do.
I'm working with matrices where every row is the same length. For example, in this $8\times 8$ every row is a vector of magnitude $\sqrt8$.
$$\begin{bmatrix} -1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 \\ 1 & -1 & 1 & 1 & 1 & -1 & 1 & -1 \\ -1 & -1 & -1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & 1 & -1 & -1 & 1 & 1 \\ 1 & -1 & 1 & 1 & -1 & 1 & -1 & 1 \\ -1 & -1 & 1 & 1 & 1 & -1 & 1 & 1 \\ -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 \\ \end{bmatrix}$$
If such a square nxn matrix had the property that every vector is orthogonal to every other vector, then that matrix would span its dimension space perfectly. Sets of vectors that aren't completely orthogonal can still span their space, but the may not do it very well. For example, $\{5, 6\}$ and $\{6, 5\}$ span $\mathbb{R}^2$, but since they're only $10$ degrees apart, they don't do it very well. On the other hand $\{5, 6\}$ and $\{5, -6\}$ are $100$ degrees apart, so those vectors do a better job of spanning $\mathbb{R}^2$.
In $2$ dimensions, this is an easy metric. The lower the dot product, the better the span, but how can I calculate this quality in higher dimensions? Does this quality have a name? How can it be computed for large matrices, such as $24\times 24$?
I tried finding the max dot product between any two pairs of vectors. This seems insufficient because I can find non reducible matrices where every pair of vectors is quite close to orthogonal.