In this paper we first prove a characterization formula for biharmonic maps in Euclidean spheres and, as an application, we construct a family of biharmonic maps from a flat 2-dimensional torus T\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {T}}$$\end{document} into the 3-dimensional unit Euclidean sphere S3\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {S}}^3$$\end{document}. Then, for the special case of maps between spheres whose components are given by homogeneous polynomials of the same degree, we find a more specific form for their bitension field. Further, we apply this formula to the case when the degree is 2, and we obtain the classification of all proper biharmonic quadratic forms from S1\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathbb {S}}}}^1$$\end{document} to Sn\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathbb {S}}}}^n$$\end{document}, n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n \ge 2$$\end{document}, from Sm\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathbb {S}}}}^m$$\end{document} to S2\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathbb {S}}}}^2$$\end{document}, m≥2\documentclass[12pt]{minimal}
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\begin{document}$$m \ge 2$$\end{document}, and from Sm\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathbb {S}}}}^m$$\end{document} to S3\documentclass[12pt]{minimal}
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\begin{document}$${{{\mathbb {S}}}}^3$$\end{document}, m≥2\documentclass[12pt]{minimal}
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\begin{document}$$m \ge 2$$\end{document}.