2. Place $n^{2}$ dominoes of size $1 \times 2$ or $2 \times 1$ on a $2 n \times 2 n$ grid without overlapping, such that each $2 \times 2$ subgrid contains at least two uncovered cells, and these two cells are in the same row or column. Find the number of all possible arrangements that meet the requirement.
2. Divide the grid into $n^{2}$ sub-grids of size $2 \times 2$, each of which can be covered by at most two dominoes. Since the dominoes exactly cover $2 n^{2}$ cells, each sub-grid must have exactly two cells covered, and these two cells must be in the same row or column.

Next, we show that these two cells are covered by the same domino.

Assume that some $2 \times 2$ sub-grids do not meet this condition. Without loss of generality, suppose there are some dominoes that cover cells in adjacent left and right sub-grids. Consider the leftmost sub-grid among such dominoes, which leads to a contradiction.

Now consider an $n \times n$ grid where each cell is a $2 \times 2$ sub-grid. Let $A, B, C, D$ represent the different scenarios for each cell, where the shaded part indicates a cell covered by a domino, as shown in Figure 2.

Fill the $n \times n$ grid with $A, B, C, D$ such that the result meets the problem's conditions, as shown in Figure 3.

From Figure 4, we know that a cell with $A$ below it (or a cell with $B$ to its right) must contain another $A$ (or $B$). Therefore, the region containing $A$ and $B$ (which may be empty) is located in the bottom-right of the grid and can be completely separated from the region containing $C$ and $D$ by a path from the bottom-left to the top-right, where the path moves one cell up or to the right each time.

Similarly, the region containing $A$ and $D$ (which may be empty) can be completely separated from the region containing $B$ and $C$ by a path from the top-left to the bottom-right, where the path moves one cell down or to the right each time.

Thus, the $n \times n$ grid is divided into four regions (which may be empty) by two paths, such that each region contains all $A, B, C, D$. Conversely, choosing two paths to divide the grid into four regions and filling them in a counterclockwise direction from the bottom with $A, B, C, D$ results in a valid domino placement.

Since each path has $\mathrm{C}_{2 n}^{n}$ ways to be chosen, the total number of domino placements is $\left(\mathrm{C}_{2 n}^{n}\right)^{2}$.