Let $n (\geq 2)$ be a positive integer. We aim to find the minimum $m$ such that there exists $x_{ij}$ (for $1 \leq i, j \leq n$) satisfying the following conditions:
1. For every $1 \leq i, j \leq n$, $x_{ij} = \max \{ x_{i1}, x_{i2}, \ldots, x_{ij} \}$ or $x_{ij} = \max \{ x_{1j}, x_{2j}, \ldots, x_{ij} \}$.
2. For every $1 \leq i \leq n$, there are at most $m$ indices $k$ such that $x_{ik} = \max \{ x_{i1}, x_{i2}, \ldots, x_{ik} \}$.
3. For every $1 \leq j \leq n$, there are at most $m$ indices $k$ such that $x_{kj} = \max \{ x_{1j}, x_{2j}, \ldots, x_{kj} \}$.

To solve this, we need to consider the structure and constraints given by the problem. The solution involves ensuring that the maximum number of indices $k$ for which $x_{ik}$ or $x_{kj}$ is the maximum is minimized.

By analyzing the constraints and constructing examples, it can be shown that the minimum $m$ satisfying the conditions is:
$$m = 1 + \left\lceil \frac{n}{2} \right\rceil.$$

Thus, the minimum value of $m$ is:
$$\boxed{1 + \left\lceil \frac{n}{2} \right\rceil}.$$