You can file with marital status called Married Filing Separately.
Then your wife has to do the same, but it is totally her responsibility.
Of course if you had most of the income you may end up paying more tax since you cannot then claim her as a dependent.
If you are unable to provide your spouse's Social Security number, you will have to file separately by mail.
If you have a past year's tax return you may be able to obtain the SS number for your spouse. If not, then you will have to file by mail. When printing, ignore the error messages related to the lack of a SS number. Include a cover letter explaining the lack of a SS number for your spouse.
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If you file MFS both of you are required by federal law to provide each other your SSN's. If your ex will not provide theirs, then have your lawyer handle it for you. That's what you paid them for. Once they explain the legalities of the requirement to provide, it shouldn't be a problem to get their SSN.