Based on my computation, I have formulated the following conjecture.
Conjecture. For any positive integer $n$, we have $$\det[|j-k|]_{1\le j,k\le n}=(-1)^{n-1}(n-1)2^{n-2}\tag{1}$$ and $$\det[|j^2-k^2|]_{1\le j,k\le n}=(-1)^{n-1}(2n-1)!!(n^2-1)2^{n-2}.\tag{2}$$
QUESTION. Are the identities $(1)$ and $(2)$ known? How to prove them?
This question looks not so difficult. Your comments are welcome!