It's possible. If you're lucky, it might be a side effect of the particular SSRI you're taking rather than the whole class of drugs. Some doctors are willing to switch you to a different SSRI or SSNRI if it looks like there's an issue, so it's worth asking. I have some experience of this:
The first one I was put on didn't have much effect. The second I insisted on being taken off after three weeks because I'd suddenly became angry all the time, with a wierd taste of blood in my mouth too. The third one hit the spot. I'm still not sure if it's working as an antidepressant, or just that it makes me able to sleep more than three hours a night.
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You're so vain
I bet you think this sig is about you