Certainly just knowing one eigenvector is not enough. If you knew three independent eigenvectors, then you would know that, using those eigenvectors as basis, the matrix is

and you could then use the known eigenvectors to write A in the standard basis.

But since -1 is a double eigenvalue, it is quite possible that there are not two independent eigenvectors for the eigenvalue -1. In that case, the matrix could be written in Jordan Normal Form, as

in

**some** basis but knowing only two eigenvectors would not give you the matrix in the standard basis.