Here by «algorithm» I mean a (halting) Turing machine with finite alphabet and memory.

Is it possible to obtain by purely existential (i.e. non-constructive) means the existence of an algorithm performing some required task, while beeing unable to explicit it (i.e. write it down or describe it constructively) ?

I hope the question is neither too vague nor too ill-posed, as I'm not so acquainted with these concepts! Thank you for any insight on the matter, feel free to retag or plainly close this naïve question if necessary.

Edit: in view of the answers given so far I think I can better explain what I meant originally. My question is more along the lines of

Is there a task for which an algorithm is known to exist while at the same time it is not possible to recognize the said algorithm?

By this I mean that our innability to perform the recognition is not a consequence of our current lack of actual knowledge, but of an intrinsec formal impossibility to do so. In other word working out that any algorithm outputs what is expected by the task is undecidable by inspection of its content.