No. Check for yourself the following fact: every outer product of vectors is a rank one matrix. Indeed, as the notation would suggest it is precisely the rank 1 matrix which takes (if we normalize everything) the first vector to the second.
A more interesting question you can ask is, is it true that every rank 1 matrix is the outer product of two vectors? Think about it this way: the image is one dimensional, and is given by the span of a vector which will have to be your ket vector (since when applying |w><v| to |u> you get <v|u> × |w>). Note that we can take this vector to be the first column of our matrix.
Now try to find a bra vector which makes your orginal matrix when you outer product with the ket vector. Hint: is the kth column a multiple of the first column? Why?
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