For analysis I would recommend using GC-MS and an Agilent DB-23 column which is able to separate cis and trans fatty acid isomers really well. You'll want to derivatitize your fatty acids to fatty acid methyl esters so they will be volatile. Your extraction protocol will depend on what fraction of the blood you want to analyze. If your analysis is from serum or plasma, a 2:1 chloroform methanol mixture should do a nice job of separating the lipid fraction. If your target is red blood cells, then you want to use the modified Rosen Folch protocol which is a combination of chloroform, methanol, and isopropanol, and has longer incubation times. I have a few papers on my research gate that outline these methods or have links to the original method paper.
For GCMS method using DB-23 column, please see "Evaluation of fatty acid and mineral content of Tanzanian seeds and oils"
For modified Rosen Folch extraction, please see the plasma phospholipid extraction in "Plasma phospholipids, non-esterified plasma polyunsaturated fatty acids and oxylipids are associated with BMI". You won't need to do the solid phase extraction unless you are interested in a particular lipid fraction such as phospholipids (in the paper).
We have a new paper coming out in Nature Scientific Reports using a rapid extraction method modified from the Water application note attached. I'd be happy to send it when it is available.
Feel free to contact directly for more help if needed.