As far as I know, multiple tests are made before organ transplant to determine matching. Would it be possible to do the matching based on the DNA of the patients, rather than the actual serum antigen/antibody? If yes, is it any better/cheaper/easier?
The better answers are being sought in a field called Regenerative Medicine.
A number of techniques have been tried, among them 3D printing of organs and dissolving the cells from a donor organ leaving the collagen scaffolding, then reseeding the organ with a patients stem cells in a bioreactor.
Most of these technologies, unfortunately, are in the proverbial 20 years from now category, which means that there are an incredible number of technical and biological challenges to overcome, but the ideal result would be organs with the patient's own cells and therefore no rejection or need for immunosuppressant therapies post transplant.
A nobel prize was awarded a few years back to Shinya Yamanaka for creating Human Induced Pluripotent Stem Cells. A technology like this would be the likely source of the stem cells necessary to seed these organs, however there have been problems with these cell lines including their tendencies to become neoplastic and form cancers.
So for at least the near and medium term the best alternative is to find organs that are as close a match as possible for Major Histocompatibility Complexes, and as Moderator Chris said, the best way to do this matching is to test for the cell surface proteins, as these are what will be the targets of immune cells. Also an organ has a very short life span after the donor dies to be usable. Sequencing is much slower than the protein screening that is in place today. You run into ethical questions regarding keeping someone on life support for the sole purpose of harvesting their organs.
And we still wouldn't be 100 percent sure that even people with the same alleles for these molecules express them in the exact same way as there may be slight differences in the post translational modifications that are made. The human immune system has evolved to detect small variations and as such we would still need to test the proteins, even if we found allelic matches. People get autoimmune diseases all the time, so there is no guarantee that even allelic matches would not be rejected.
The human genome is 3 billion letters long. Most of that sequence has nothing to do with tissue rejection. Only a handful of genes relate to tissue rejection, so those are what one might want to sequence, but even having the sequence doesn't necessarily tell you how the proteins are shaped, and it's protein shape that determines how they interact with each other.