Consider a lattice in R^3. Is the some "canonical" way or ways to choose basis in it ?

I mean in R^2 we can choose a basis |h_1| < |h_2| and |(h_2, h_1)| < 1/2 |h_1|. Considering lattices with fixed determinant and up to unitary transformations we get standard picture of the PSL(2,Z) acting on the upper half plane, which has a fundamental domain Im (tau)>1 Re(tau) <1/2.

What are the similar results for other small dimensions R^3, R^4, C^4, C^8 ? What are the algorithms to find such a lattice reductions ?