This paper presents two new computationally efficient direct methods for fitting n-dimensional ellipsoids to noisy data. They conduct the fitting by minimizing the algebraic distance in subject to suitable quadratic constraints. The hyperellipsoid-specific (HES) method is an elaboration of existing ellipse and 3D ellipsoid-specific fitting methods. It is shown that HES is ellipsoid-specific in n-dimensional space. A limitation of HES is that it may provide biased fitting results with data originating from an ellipsoid with a large ratio between the longest and shortest main axis. The sum-of-discriminants (SOD) method does not have such a limitation. The constraint used by SOD rejects a subset of non-ellipsoidal quadrics, which enables a high tendency to produce ellipsoidal solutions. Moreover, a regularization technique is presented to force the solutions towards ellipsoids with SOD. The regularization technique is compatible also with several existing 2D and 3D fitting methods. The new methods are compared through extensive numerical experiments with n-dimensional variants of three commonly used direct fitting approaches for quadratic surfaces. The results of the experiments imply that in addition to the superior capability to create ellipsoidal solutions, the estimation accuracy of the new methods is better or equal to that of the reference approaches.