  Let $V$ be a $n$-dimensional real vector space. In this paper we introduce
the concept of \emph{euclidean} Clifford algebra $\mathcal{C\ell}(V,G_{E})$ for
a given euclidean structure on $V,$ i.e., a pair $(V,G_{E})$ where $G_{E}$ is a
euclidean metric for $V$ (also called an euclidean scalar product). Our
construction of $\mathcal{C\ell}(V,G_{E})$ has been designed to produce a
powerful computational tool. We start introducing the concept of
\emph{multivectors} over $V.$ These objects are elements of a linear space over
the real field, denoted by $\bigwedge V.$ We introduce moreover, the concepts
of exterior and euclidean scalar product of multivectors. This permits the
introduction of two \emph{contraction operators} on $\bigwedge V,$ and the
concept of euclidean \emph{interior} algebras. Equipped with these notions an
euclidean Clifford product is easily introduced. We worked out with
considerable details several important identities and useful formulas, to help
the reader to develope a skill on the subject, preparing himself for the
reading of the following papers in this series.
