If M M is a complete minimal surface in R n {R^n} , we denote by W W the set of points in R n {R^n} that do not lie on any tangent plane of M M . By taking a point in W W as origin, the position vector of M M determines a global unit normal vector field e e to M M . We prove that if e e is a minimal section, then M M is a plane. In particular, the set of tangent planes of a nonflat complete minimal surface in R 3 {R^3} covers all R 3 {R^3} . We also prove a similar result for a complete minimal surface M M in S 3 {S^3} , and deduce from it that if the spherical image of M M lies in a closed hemisphere, then M M is a great S 2 {S^2} .