How does one construct an abstract (n × n) empty grid of squares?

Question

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I would like to just create a grid like this, but without numbers or color, but I'd like to represent an n × n grid abstractly with ellipses like this:

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Also, I need just a simple—singular—grid like that of the top left grid in the image above.

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I know need a red line (y=x) from the lower left corner to the upper right corner.

Answer 1

A very basic approach using grid:

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\draw (0,0) grid (3,3);
\draw (0,4) grid (3,7);
\draw (4,0) grid (7,3);
\draw (4,4) grid (7,7);
\foreach \i/\valor in {1/1,2/2,3/3,5/n-2,6/n-1,7/n}
  {
    \node[anchor=south] at (\i-0.5,7) {$\valor$};
    \node[anchor=east] at (0,-\i+7.5) {$\valor$};
}
\node at (3.5,1.5) {$\cdots$};
\node at (3.5,5.5) {$\cdots$};
\node at (1.5,3.5) {$\vdots$};
\node at (5.5,3.5) {$\vdots$};
\node at (3.5,3.5) {$\ddots$};
\end{tikzpicture}

\end{document}

With the new requirement of the edited question:

\documentclass{article}
\usepackage{tikz}

\begin{document}

\begin{tikzpicture}
\draw (0,0) grid (3,3);
\draw (0,4) grid (3,7);
\draw (4,0) grid (7,3);
\draw (4,4) grid (7,7);
\foreach \i/\valor in {1/1,2/2,3/3,5/n-2,6/n-1,7/n}
  {
    \node[anchor=south] at (\i-0.5,7) {$\valor$};
    \node[anchor=east] at (0,-\i+7.5) {$\valor$};
}
\node at (3.5,1.5) {$\cdots$};
\node at (3.5,5.5) {$\cdots$};
\node at (1.5,3.5) {$\vdots$};
\node at (5.5,3.5) {$\vdots$};
\node at (3.5,3.5) {$\ddots$};
\draw[red,ultra thick] (0,0) -- (7,7);
\end{tikzpicture}

\end{document}