The formula for  dimensions can be derived assuming an -dimensional real vector space  with a basis  and an inner product . The reciprocal lattice vectors are uniquely determined by the formula . Using the permutation
they can be determined with the following formula:
Here,  is the volume form,  is the inverse of the vector space isomorphism  defined by  and  denotes the inner multiplication.
One can verify that this formula is equivalent to the known formulas for the two- and three-dimensional case by using the following facts: In three dimensions,  and in two dimensions, , where  is the rotation by 90 degrees (just like the volume form, the angle assigned to a rotation depends on the choice of orientation).