Matrix form of 2D-DFT for a vectorized image
In matrix form, we can write the 2-D DFT of an Image $\mathbf X$ via $\hat{\mathbf{X}} = \mathbf F \mathbf X \mathbf F^{\rm T}$.

This expression can be vectorized with the help of the Kronecker product. Using [*], we obtain $$\hat{{\mathbf{x}}} = {\rm vec}\{\hat{\mathbf{X}}\} = (\mathbf F \otimes \mathbf F) \cdot \mathbf{x}.$$

This shows that the matrix you need is $\mathbf F \otimes \mathbf F$.