The only thing I know is to use gram-schmidt and see if I get back the same matrices. But I thought perhaps that's not the method you should use if all you want to do is check for orthogonality, and maybe there is a smarter way?
I am a beginner, so Gram-Schmidt is about as advanced as things go for me. Would be very grateful for any help!
There is no better way than computing all $\frac{(n-1)n}2$ inner products, as they are independent of each other.
So in principle $O(n^3)$ operations. If $n$ is large enough ($n>100$), a fast matrix product algorithm such as Strassen can be thought of. This will lower to $O(n^{\log_27})$ operations.
Gram-Schmidt is a viable alternative, as the complexity is also $O(n^3)$ [$n^3$ additions and multiplications; also $n^2$ divisions]; be sure to use the modified Gram-Schmidt version, for better numerical stability.