Vectoral and alignment-free approaches to biological sequence representation have been explored in bioinformatics to efficiently handle big data. Even so, most current methods involve sequence comparisons via alignment-based heuristics and fail when applied to the analysis of large data sets. Here, we present "Spaced Words Projection (SWeeP)", a method for representing biological sequences using relatively small vectors while preserving intersequence comparability. SWeeP uses spaced-words by scanning the sequences and generating indices to create a higher-dimensional vector that is later projected onto a smaller randomly oriented orthonormal base. We constructed phylogenetic trees for all organisms with mitochondrial and bacterial protein data in the NCBI database. SWeeP quickly built complete and accurate trees for these organisms with low computational cost. We compared SWeeP to other alignment-free methods and Sweep was 10 to 100 times quicker than the other techniques. A tool to build SWeeP vectors is available at https://sourceforge.net/projects/spacedwordsprojection/.