I have been reading 'The Arithmetic of Elliptic Curves' by Silverman.
It proves that any elliptic curve defined over a perfect field can be reduced to Weierstrass form, [III.3.1,page 59]. The convention $k$ is perfect is mentioned at the begining of the book.
The proof uses the fact that $\bar{k}/k$ is Galois when $k$ is perfect.
Hence I don't know how to move the proof to the case $k$ is non-perfect.
The answer is yes. And the result can be found in J.S.Milnes’ book, Elliptic Curves, II, theorem 2.4, page 53. And it starts with “Let $k$ be an arbitrary field,” which is exactly what I want.
It depends on the Riemann-roch theorem over arbitrary field, proved in QingLiu's book, Algebraic Geometry and Arithmetic Curve, 7.3.2, Remark 3.33. See also 7.4.1, Corollary 4.5 which gives a closed embedding of elliptic curve into $\mathbb{P}^2_k$.