Check out the SILVA database — in addition to the SSU database that is most commonly used, they have a LSU database, which would include 23S. What I do not know is if it is comprehensive enough for your purposes (e.g., has all the species you expect to see). I'd start there — building on an existing database is always much easier than building a new one from scratch!
what do the header lines look like in that fasta? If it is something like:
>sequenceID| d_Eukaryota;k_Chromista;p_Ochrophyta;c_Phaeophyceae;o_Fucales;f_Hormosiraceae;g_Hormosira;s_Hormosira_banksii
ACGTGTAGTGTGCTGTAGTAGTCGTGAC
then you can easily do this with some bash commands. Something like this:
grep '>' pathto.fasta | tr -d '>' | tr '|' '\t'
But if the full taxonomy string is not in the header line of the fasta then that will not help.