Painting An $n\times n$ Grid

One solution for $n=25$ is when you paint 25 5x5 grids, each a uniform color, and then arrange them in 5 rows of 5. The top of this solution looks like this:

Now this can't be done for $n>25$ , and here is my attempt at a proof by contradiction:

Assume you can do it for $n=26$ . Then you add up the total number of colors used in each rows and the total will be, at most , (5 colors in each row/column)x(52 total rows and columns) = 260.

However, it is easy to see that however you color 26 squares, they will occupy at least 11 total number of rows and columns. So, the minimum number of total colors is given by $26\times 11 = 286$ .

This would imply that $286 \leq 260$ , which is a contradiction. So, there would be no way to do it for a 26x26 matrix. It can be similarly shown for any number greater than 26 as well.