It depends if you are doing this on the solid phase or in solution. This can be easily accomplished on a DNA synthesizer by using a commercially available "spacer" phosphoramidite to separate two oligonucleotide fragments. On the other hand, if you want to try this in solution, you will need to modify your oligonucleotides with reactive groups. One way is to synthesize amino-modified oligos and then join them using a homobifunctional PEG NHS ester, if the oligos are the same. If the two oligonucleotides have different sequences, I would suggest synthesizing one with an amine and the other with a terminal or strained alkyne and then joining them using a heterobifunctional PEG linker (NHS/azide).
The two DNA srands I want to link are different in sequence, and the idea you suggested can be a good candidate. But it really puzzles me how to perform chemical synthesing and linking as I am new in this area. Can you provide more detailed experimental protocols ? Thanks very much!
Generally, you will find the necessary protocolls to do crosslinking in the paperwork delivered with the crosslinkers. If you for example get a heterobifunctional crosslinker from Pierce (Thermo Fisher), they are providing conditions usually applied for such reactions. You can download the procedures even without buying.
Pierce (piercenet.com) provides quite large knowledge base for crosslinking at their website (https://www.thermofisher.com/de/de/home/life-science/protein-biology/protein-labeling-crosslinking/protein-crosslinking.html) the link is for Germany, but you will find your way for the english version. The protocolls are mostly set up for Proteins, but will work for DNA as well if the DNA is equipped the necessary reactive groups.
For chemical DNA Synthesis go for LinkTechnologies website and get their catalogue. It is an excellent guide to oligonucleotide chemistry and modification.