$(A^T)^{-1}=(A^{-1})^T$ and according to Wikipedia, a skew-symmetric matrix is a matrix that satisfies the condition $A^T=-A$. So $(A^{-1})^T=(A^T)^{-1}=(-A)^{-1}=-A^{-1}$
Why do you need $2n\times 2n$ condition?
Consider what happens when you're getting the matrix of minors for a $3\times 3$: You are getting the value for the $a_{ij}$ element by calculating the determinant of the non $i,j$ rows (I won't go through this since, judging by your question, you know how to do this already).
Now consider the same logic applied to a $2\times 2$ matrix: Do the same steps, except instead of having a $2\times 2$ matrix to calculate the determinant from, you have a $1\times 1$. In this case, the determinant is the single element in that matrix. From this, you can do the same steps as you would for a $3\times 3$.
I'll run through an example here (I'll compare with a $3\times 3$ matrix since that's the simplest matrix where these rules are first introduced):
Matrix $A = \begin{bmatrix} a & b\\ c & d\end{bmatrix}$.
Matrix of minors $= \begin{bmatrix} d & c\\ b & a\end{bmatrix}$ - for element $(1,1)$ (i.e. $a$), "cover" row $1$ and column $1$ and you're left with $d$ so that equals the determinant for that element for the matrix of minors (as with a $3\times 3$).
Cofactor matrix, $C = \begin{bmatrix} d & -c\\ -b & a\end{bmatrix}$ - exact same as for a $3\times 3$.
$\operatorname{adj}A = C^T = \begin{bmatrix} d & -b\\ -c & a\end{bmatrix}$ - again, exact same as for a $3\times 3$.
This is where you get the "Just remember it" from. It follows the exact same steps as for any $n\times n$ matrix.
Then you calculate $A^{-1}$ as usual: $A^{-1} = \frac{1}{\det A}\operatorname{adj}A$.
Hope this helps.
Best Answer
Suppose that all entries of an invertible matrix $A$ are real numbers. Then its determinant is real (and not $0$) and all entries of the adjugate matrix of $A$ are real numbers (its entries are determinants of real matrices). So, $A^{-1}$ is a real matrix too.
Now, apply this to $C^{-1}$: if it was a real matrix, $C$ would be a real matrix too.
By the way, this works for every field.