Representation learning is an important step in the machine learning pipeline. Given the current biological sequencing data volume, learning an explicit representation is prohibitive due to the dimensionality of the resulting feature vectors. Kernel-based methods, e.g., SVM, are a proven efficient and useful alternative for several machine learning (ML) tasks such as sequence classification. Three challenges with kernel methods are (i) the computation time, (ii) the memory usage (storing an $n\times n$ matrix), and (iii) the usage of kernel matrices limited to kernel-based ML methods (difficult to generalize on non-kernel classifiers). While (i) can be solved using approximate methods, challenge (ii) remains for typical kernel methods. Similarly, although non-kernel-based ML methods can be applied to kernel matrices by extracting principal components (kernel PCA), it may result in information loss, while being computationally expensive. In this paper, we propose a general-purpose representation learning approach that embodies kernel methods' qualities while avoiding computation, memory, and generalizability challenges. This involves computing a low-dimensional embedding of each sequence, using random projections of its $k$-mer frequency vectors, significantly reducing the computation needed to compute the dot product and the memory needed to store the resulting representation. Our proposed fast and alignment-free embedding method can be used as input to any distance (e.g., $k$ nearest neighbors) and non-distance (e.g., decision tree) based ML method for classification and clustering tasks. Using different forms of biological sequences as input, we perform a variety of real-world classification tasks, such as SARS-CoV-2 lineage and gene family classification, outperforming several state-of-the-art embedding and kernel methods in predictive performance.