By "corresponding submatrices" I presume you mean those $2\times2$ minors obtained by deleting $n-2$ colums and $n-2$ rows, where these columns and rows have the same $n-2$ indices. Once you've calculated the determinants of these submatrices you recover the action of $A$ on the exterior square $\Lambda^2 V$.

Now the following paper: "An algorithm for recognising the exterior square of a matrix" by Catherine Greenhill explains how to then obtain the original matrix $A$. Here's the relevant quote:

> One computational problem which presents itself immediately is this: how can we
determine whether a given matrix $Y$ is equal to the exterior square of another matrix $X$?
In particular, if such an $X$ exists then we would like to construct one. A polynomial-time
algorithm which solves this problem is described in Section 5.

The paper can be downloaded [here][1].

One slight problem is that the exterior square does not quite determine the matrix $X$ uniquely. Here is another quote from the paper:

> We prove in Section 4 that two matrices $X$, $X'$ with rank at least three
have the same exterior square if and only $X'\in \{X, -X\}$.

So if the rank is at least three, then we are pretty much done. When the rank is less than three I'm not sure what happens - you will need to make use of other $2\times 2$ minors than the ones described above...


  [1]: http://web.maths.unsw.edu.au/~csg/papers/ext-matrix.pdf