1. Coloring the Board:
We start by coloring the $5 \times 5$ board in a specific pattern. Paint the corners, the center square, and the middle squares of each edge black. This results in the following pattern:
$$\begin{array}{|c|c|c|c|c|} \hline B & W & W & W & B \\ \hline W & W & W & W & W \\ \hline W & W & B & W & W \\ \hline W & W & W & W & W \\ \hline B & W & W & W & B \\ \hline \end{array}$$
Here, $B$ represents a black square and $W$ represents a white square.

2. Analyzing the Hook Shape:
Each hook (a three-unit square figure) covers exactly one black square and two white squares. This is because the hook shape can be visualized as:
$$\begin{array}{|c|c|} \hline B & W \\ \hline W & W \\ \hline \end{array}$$
or any rotation of this shape.

3. Counting Black Squares:
There are 9 black squares on the board (4 corners, 4 middle of edges, and 1 center).

4. Covering the Board:
Since each hook covers exactly one black square, and we have 8 hooks, we can cover 8 black squares. This leaves exactly one black square uncovered.

5. Conclusion:
The only squares that can remain free are the black squares. Therefore, the possible positions for the free square are:
- The four corners: (1,1), (1,5), (5,1), (5,5)
- The center of the edges: (1,3), (3,1), (3,5), (5,3)
- The center of the board: (3,3)

The final answer is \{(1,1), (1,5), (5,1), (5,5), (1,3), (3,1), (3,5), (5,3), (3,3)\}.