Homologous chromosomes do not pair during mitosis, so there is no opportunity for crossing over to occur. Crossing over between non-sister chromatids of homologous chromosomes occurs in meiosis during prophase I.
During mitosis, chromosomes duplicate to result in identical sister chromatids in prophase. These then align in the middle of the cell during metaphase and are pulled apart by spindle fibers during anaphase and telophase to arrive at opposite poles of the cell. The cells are then disjointed, resulting in two identical daughter cells. Since this replication is meant to result in identical cells, non-sister chromatids are never formed, and crossing over cannot occur. Crossing over does not occur within the single chromosome during mitosis but between two single sister chromatids from two different chromosomes during meiosis.
Before prophase I of meiosis, the oogonium contains diploid chromosomes but tetraploid DNA, with each chromosome being made of two sister chromatids. During prophase I, the two chromosomes align (two sets of identical sister chromatids), and crossing over occurs at a chiasma between two non-sister chromatids. The alignment of these two pairs of chromosomes is necessary for crossing over to occur, whereas in mitosis, there is only the alignment and subsequent dissection of single duplicated chromosomes.