In the following, we define infinite Hadamard matrices.

Let $\omega$ be the set of non-negative integers. If $f,g:\omega\to\{-1,1\}$ are maps, then we say $f,g$ are approximately orthogonal if $$\lim\sup_{n\to\infty}\sum_{k=0}^n \big(f(k)\cdot g(k)\big) = 0 = \lim\inf_{n\to\infty}\sum_{k=0}^n \big(f(k)\cdot g(k)\big).$$

An infinite Hadamard matrix is a map $M:\omega^2 \to \{-1,1\}$ such that whenever $i\neq j \in\omega$ then the "row vectors" $M[i,\cdot]$ and $M[j,\cdot]$ are approximately orthogonal. (Given $a\in\omega$, the map $M[a,\cdot]:\omega\to\{-1,1\}$ is defined by $n\mapsto M(a, n)$.)

Question. What is an example of an infinite Hadamard matrix?