If I want to rotate the standard coordinate system with axis {0,0,1} {0,1,0} {1,0,0} to another coordinate system with axis say vectors A, B ,C, how can I get the rotation matrix?

If A, B and C form an orthonormal basis, then your rotation matrix is simply formed by the basis vector A, B, C:

Matrix3f rot;
rot << A, B, C;

Here rot is the transformation from the local space (A,B,C) to world coordinates. To get he opposite rotation, simply take or construct the transpose of rot.